Solving Nonlinear Functions - PSAT Math

1. Solve for $x$: $(x-4)(x+1)=0$.

   $x=4$ or $x=-1$. Set each factor to zero: $x-4=0$ or $x+1=0$.

2. What does $\Delta=0$ imply about the real solutions of $ax^2+bx+c=0$?

   $1$ real solution (a repeated root). Zero discriminant makes $\pm\sqrt{0}=0$, giving one solution.

3. What is the standard method to solve $a(bx+c)^2=d$ when $a\ne 0$ and $d/a\ge 0$?

   $(bx+c)=\pm\sqrt{\frac{d}{a}}$. Isolate the squared term, then take square root of both sides.

4. Identify the solutions of $x^4-16=0$.

   $x=\pm 2$. Factor as $(x^2-4)(x^2+4)=0$; only $x^2=4$ has real solutions.

5. Identify the first step to solve $f(x)=g(x)$ when $f$ and $g$ are nonlinear functions.

   Set $f(x)-g(x)=0$ and solve the resulting equation. Rearranging creates a single equation to solve.

6. State the quadratic formula for solutions of $ax^2+bx+c=0$.

   $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the standard form.

7. What is the axis of symmetry for $y=a(x-h)^2+k$?

   $x=h$. The vertical line through the vertex where the parabola is symmetric.

8. What is the solution set of $\sqrt{x-1}=3$?

   $x=10$. Square both sides: $x-1=9$, so $x=10$.

9. Solve for $x$: $(x+1)^2=16$.

   $x=3$ or $x=-5$. Take square root: $x+1=\pm 4$, so $x=3$ or $x=-5$.

10. State the vertex $x$-coordinate formula for $y=ax^2+bx+c$.

    $x=-\frac{b}{2a}$. The parabola's vertex occurs where the derivative equals zero.

11. What are the solutions to $x^2+6x+9=0$?

    $x=-3$. Perfect square $(x+3)^2=0$ has one repeated root.

12. What is the zero-product property used after factoring?

    If $ab=0$, then $a=0$ or $b=0$. At least one factor must be zero for the product to be zero.

13. What is the quadratic formula for solutions to $ax^2+bx+c=0$?

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived from completing the square on the general quadratic equation.

14. What is the zero-product property used when solving factored equations?

    If $ab=0$, then $a=0$ or $b=0$. At least one factor must be zero for the product to be zero.

15. What are the solutions to $x^2-9=0$?

    $x=\pm 3$. Factor as $(x-3)(x+3)=0$, then apply zero-product property.

16. Solve for $x$: $x^4-5x^2+4=0$.

    $x=\pm 1$ or $x=\pm 2$. Let $u=x^2$: $u^2-5u+4=0$ factors to $(u-1)(u-4)=0$.

17. What substitution turns a quadratic in $x^2$ into a linear equation in $u$?

    Let $u=x^2$. This transforms equations like $x^4+3x^2-4=0$ into $u^2+3u-4=0$.

18. What are the solutions to $2x^2-8=0$?

    $x=\pm 2$. Divide by 2: $x^2=4$, then take square roots.

19. What are the solutions to $x^2+5x=0$?

    $x=0$ or $x=-5$. Factor out $x$: $x(x+5)=0$, then apply zero-product property.

20. What does $b^2-4ac=0$ tell you about the real solutions of $ax^2+bx+c=0$?

    One real solution (a double root). Zero discriminant makes $\pm\sqrt{0}=0$, so both roots coincide.

21. What is the solution set of $\sqrt{x+1}=3$?

    $x=8$. Square both sides: $x+1=9$, so $x=8$.

22. What is the square root property for $(x-h)^2=k$?

    $x=h\pm\sqrt{k}$. Take square root of both sides, giving two solutions.

23. What does $\Delta>0$ indicate about the real solutions of $ax^2+bx+c=0$?

    Two distinct real solutions. Positive discriminant means the square root yields two different values.

24. What is the solution set of $x^2=-16$ over the real numbers?

    No real solution. Square of real number cannot be negative.

25. Solve for $x$: $\sqrt{x-1}=5$.

    $x=26$. Square both sides: $x-1=25$, so $x=26$.

26. What does $\Delta=b^2-4ac=0$ imply about the solutions of $ax^2+bx+c=0$?

    $1$ real double solution. Zero discriminant means both roots coincide.

27. What are the real solutions to $(x-2)^2=9$?

    $x=5$ or $x=-1$. Take square root: $x-2=\pm 3$, so $x=5$ or $x=-1$.

28. What are the solutions to $x^2-5x=0$?

    $x=0$ or $x=5$. Factor as $x(x-5)=0$, giving two solutions.

29. What is the solution set of $\sqrt{x-1}=3$?

    $x=10$. Square both sides: $x-1=9$, so $x=10$.

30. What is the axis of symmetry of $y=a(x-h)^2+k$?

    $x=h$. In vertex form, the parabola is symmetric about the vertical line $x=h$.

31. What does $\Delta<0$ imply about the real solutions of $ax^2+bx+c=0$?

    No real solutions. Negative discriminant means no real square root exists.

32. What is the zero-product property used when factoring nonlinear equations?

    If $ab=0$, then $a=0$ or $b=0$. Allows solving factored equations by setting each factor to zero.

33. What step is required after squaring both sides of an equation with radicals?

    Check for extraneous solutions. Squaring can introduce false solutions that don't satisfy the original.

34. Identify the key step to solve $x^2=k$ for real $x$ when $k\ge 0$.

    $x=\pm\sqrt{k}$. Take square root of both sides, considering both positive and negative roots.

35. What are the solutions of $x^2-5x+6=0$?

    $x=2$ and $x=3$. Factors to $(x-2)(x-3)=0$; apply zero-product property.

36. What is the discriminant of $ax^2+bx+c=0$?

    $b^2-4ac$. The expression under the square root in the quadratic formula.

37. What are the solutions to $x^2=\frac{1}{4}$?

    $x=\pm\frac{1}{2}$. Take square root of both sides: $x=\pm\sqrt{\frac{1}{4}}$.

38. What is the solution set of $|x-3|=5$?

    $x=8$ and $x=-2$. Split into two cases: $x-3=5$ or $x-3=-5$.

39. What is the solution of $x^2=50$ in simplest radical form?

    $x=\pm 5\sqrt{2}$. Take square root of both sides: $x=\pm\sqrt{50}=\pm\sqrt{25\cdot 2}=\pm 5\sqrt{2}$.

40. What is the standard form of a quadratic function?

    $y=ax^2+bx+c$. General polynomial form with degree 2; coefficients determine shape.

41. Identify the vertex of $y=(x-2)^2+5$.

    $(2,5)$. In vertex form $(x-h)^2+k$, the vertex is at $(h,k)$.

42. State the quadratic formula for solutions of $ax^2+bx+c=0$.

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the standard form.

43. What is the key restriction when solving a rational equation with denominators in $x$?

    Exclude values that make any denominator equal to $0$. Division by zero is undefined, so these values must be excluded.

44. What is the discriminant for $ax^2+bx+c=0$, and what does its sign determine?

    $\Delta=b^2-4ac$; sign gives number of real solutions. Positive means 2 real roots, zero means 1, negative means none.

45. What is the square root property for solving $(x-h)^2=k$?

    $x=h\pm\sqrt{k}$. Take square root of both sides, remembering $\pm$ for both solutions.

46. What are the solutions of $(x-4)(x+1)=0$?

    $x=4$ and $x=-1$. Apply zero-product property: each factor equals zero.

47. What is the standard form of a circle centered at $(h,k)$ with radius $r$?

    $(x-h)^2+(y-k)^2=r^2$. Distance formula squared: all points exactly $r$ units from $(h,k)$.

48. What is the solution set of $x^2+5x=0$?

    $x=0$ and $x=-5$. Factor out $x$: $x(x+5)=0$, so $x=0$ or $x=-5$.

49. What is the inverse operation used to solve $x^2=k$ for real $x$?

    $x=\pm\sqrt{k}$ (requires $k\ge 0$). Take square root of both sides; negative $k$ has no real solutions.

50. What is the substitution that turns a biquadratic $x^4+bx^2+c=0$ into a quadratic?

    Let $u=x^2$. Reduces the degree from 4 to 2, making it solvable by quadratic formula.

51. What domain restriction must be stated for $\frac{1}{x-3}$ before solving equations?

    $x\ne 3$. The denominator cannot be zero, so exclude $x=3$.

52. Identify the method: What is the first step to solve $(x-3)(x+5)=0$?

    Set each factor equal to $0$. Zero product property: if $AB=0$, then $A=0$ or $B=0$.

53. What is the quadratic formula for solving $ax^2+bx+c=0$ in terms of $a$, $b$, and $c$?

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived using completing the square on the general quadratic equation.

54. Identify the solutions of $\sqrt{x-1}=3$.

    $x=10$. Square both sides: $x-1=9$, so $x=10$.

55. What is the axis of symmetry for $y=ax^2+bx+c$?

    $x=-\frac{b}{2a}$. The $x$-coordinate of the vertex, halfway between roots.

56. What does $\Delta<0$ indicate about the real solutions of $ax^2+bx+c=0$?

    No real solutions. Negative discriminant means $\sqrt{\Delta}$ is imaginary.

57. What is the standard form of a quadratic equation to solve for its zeros?

    $ax^2+bx+c=0$ with $a\ne 0$. Standard form requires the equation to equal zero with $a \ne 0$.

58. What is the first step to solve an equation with a rational expression in $x$?

    Multiply both sides by the LCD. Clears denominators to create a polynomial equation.

59. What is the solution set of $\frac{1}{x}=2$?

    $x=\frac{1}{2}$. Multiply both sides by $x$: $1=2x$, so $x=\frac{1}{2}$.

60. What does $b^2-4ac<0$ tell you about the real solutions of $ax^2+bx+c=0$?

    No real solutions. Negative discriminant gives imaginary square roots, no real values.

61. What does $\Delta<0$ tell you about the real solutions of $ax^2+bx+c=0$?

    No real solutions. Negative discriminant means square root of negative number, which isn't real.

62. What is the vertex $x$-coordinate of $y=ax^2+bx+c$ in terms of $a$ and $b$?

    $x=-\frac{b}{2a}$. The vertex lies on the axis of symmetry.

63. Solve for $x$: $x^2-5x+6=0$.

    $x=2$ or $x=3$. Factors to $(x-2)(x-3)=0$ using zero-product property.

64. Identify the substitution that makes $x^4-5x^2+4=0$ quadratic in one variable.

    Let $u=x^2$. Transforms to $u^2-5u+4=0$, a standard quadratic form.

65. What is the solution to the system $y=x^2$ and $y=9$?

    $(x,y)=(3,9)$ and $(x,y)=(-3,9)$. Substitute $y=9$ into $y=x^2$: $x^2=9$, so $x=\pm 3$.

66. Solve for $x$: $x^2-9=0$.

    $x=\pm 3$. Factor as $(x-3)(x+3)=0$, giving $x=3$ or $x=-3$.

67. State the factored form of a quadratic with zeros $r_1$ and $r_2$.

    $y=a(x-r_1)(x-r_2)$. Each factor $(x-r)$ equals zero when $x$ equals that root.

68. What does $\Delta=b^2-4ac>0$ imply about the solutions of $ax^2+bx+c=0$?

    $2$ distinct real solutions. Positive discriminant means the square root is real.

69. Solve $x^2=49$ for $x$.

    $x=\pm 7$. Take square root of both sides: $x=\pm\sqrt{49}$.

70. What is the quadratic formula for solving $ax^2+bx+c=0$?

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived using completing the square on the general quadratic equation.

71. What are the solutions of $(x-2)(x+5)=0$?

    $x=2$ or $x=-5$. Apply zero-product property: each factor equals zero.

72. What is the quadratic formula for solving $ax^2+bx+c=0$?

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived from completing the square on the standard form.

73. What are the solutions to $x^2=49$?

    $x=\pm 7$. $\sqrt{49}=7$, so $x=\pm 7$.

74. Solve for $x$: $x^2=49$.

    $x=\pm 7$. Take square roots of both sides: $x=\pm\sqrt{49}=\pm 7$.

75. What is the vertex form of a quadratic function?

    $y=a(x-h)^2+k$. Shows parabola with vertex at $(h,k)$ and vertical stretch factor $a$.

76. Solve $x^2-5x=0$ for $x$.

    $x=0$ or $x=5$. Factor out $x$: $x(x-5)=0$, then use zero-product property.

77. What are the solutions to $(x-4)(x+1)=0$?

    $x=4$ or $x=-1$. Apply zero-product property to each factor.

78. What is the solution set of $\frac{2}{x-1}=1$?

    $x=3$. Cross-multiply: $2=x-1$, so $x=3$.

79. What are the solutions to $(x-4)^2=25$?

    $x=9$ or $x=-1$. Take square root: $x-4=\pm 5$, so $x=4\pm 5$.

80. Solve for $x$: $x^2+4x+5=0$.

    No real solutions. Discriminant is $16-20=-4<0$, so no real solutions exist.

81. What is the solution set of $\sqrt{x+2}=x$?

    $x=2$. Square both sides: $x+2=x^2$; solve $x^2-x-2=0$ to get $x=2$ (reject $x=-1$).

82. What are the solutions to $(x-4)(x+1)=0$?

    $x=4$ or $x=-1$. Apply zero-product property: each factor can equal zero.

83. What is the axis of symmetry of $y=ax^2+bx+c$ written in terms of $a$ and $b$?

    $x=-\frac{b}{2a}$. The parabola's line of symmetry passes through the vertex.

84. Solve for $x$: $x^2-5x=0$.

    $x=0$ or $x=5$. Factor out $x$: $x(x-5)=0$, so $x=0$ or $x=5$.

85. What are the solutions to $x^2=16$?

    $x=\pm 4$. Take square root of both sides: $x=\pm\sqrt{16}$.

86. What is the solution of $\sqrt{x-1}=4$?

    $x=17$. Square both sides: $x-1=16$, so $x=17$.

87. Solve for $x$: $3x^2-12=0$.

    $x=\pm 2$. Add 12 to both sides: $3x^2=12$, so $x^2=4$, thus $x=\pm 2$.

88. Solve for $x$: $x^{\frac{1}{2}}=3$.

    $x=9$. Square both sides: $(x^{\frac{1}{2}})^2=3^2$.

89. What does $\Delta<0$ imply about the solutions of $ax^2+bx+c=0$?

    No real solutions (two complex solutions). Negative discriminant requires imaginary numbers for the square root.

90. What is the solution to $x^2-6x+9=0$?

    $x=3$. Perfect square: $(x-3)^2=0$, so $x=3$ (double root).

91. What does $b^2-4ac>0$ tell you about the real solutions of $ax^2+bx+c=0$?

    Two distinct real solutions. Positive discriminant means the square root is real, yielding two values.

92. Solve for $x$: $\sqrt{x+1}=3$.

    $x=8$. Square both sides: $x+1=9$, so $x=8$.

93. What does it mean if $\Delta<0$ for $ax^2+bx+c=0$?

    No real solutions. Negative discriminant means the parabola doesn't cross the x-axis.

94. What does $b^2-4ac=0$ tell you about solutions of $ax^2+bx+c=0$?

    $1$ real solution (a double root). Zero discriminant means $\pm\sqrt{0}=0$, giving one repeated root.

95. Solve $\frac{1}{x-2}=3$ for $x$.

    $x=\frac{7}{3}$. Cross-multiply: $1=3(x-2)$, so $1=3x-6$.

96. Identify the $x$-intercepts of $y=a(x-r_1)(x-r_2)$.

    $x=r_1$ and $x=r_2$. The parabola crosses the $x$-axis where each factor equals zero.

97. What does $\Delta=0$ indicate about the real solutions of $ax^2+bx+c=0$?

    One real solution (a double root). Zero discriminant makes $\pm\sqrt{0}=0$, giving one repeated root.

98. What does $b^2-4ac<0$ tell you about solutions of $ax^2+bx+c=0$?

    No real solutions. Negative discriminant means square root of negative number is not real.

99. What is the zero-product property used after factoring an equation like $(x-3)(x+2)=0$?

    If $ab=0$, then $a=0$ or $b=0$. A product equals zero only when at least one factor is zero.

100. What is the standard form of a quadratic equation in $x$?

     $ax^2+bx+c=0$ with $a\ne 0$. The general form with $a$ as the coefficient of $x^2$.

101. Solve for $x$: $x^2+6x+9=0$.

     $x=-3$. This is $(x+3)^2=0$, a perfect square with one repeated root.

102. Solve for $x$: $x^4-16=0$.

     $x=\pm 2$. Factor as $(x^2-4)(x^2+4)=0$; only $x^2=4$ has real solutions.

103. Solve for $x$: $x^2+6x+9=0$.

     $x=-3$. Perfect square: $(x+3)^2=0$ gives a double root.

104. Solve $x^2-6x+9=0$ for $x$.

     $x=3$. Perfect square: $(x-3)^2=0$, so $x-3=0$.

105. What is the solution set of $x^2+1=0$ over the real numbers?

     No real solutions. $x^2=-1$ has no real square root.

106. What is the standard form of a quadratic equation in $x$?

     $ax^2+bx+c=0$ with $a\ne 0$. The coefficient $a$ must be nonzero for a quadratic equation.

107. What is the definition of a solution to $f(x)=g(x)$?

     Any $x$ where $f(x)$ and $g(x)$ are equal. The x-values where the two functions intersect.

108. What is the axis of symmetry for $y=a(x-h)^2+k$?

     $x=h$. The parabola is symmetric about the vertical line through the vertex.

109. What is the quadratic formula for solutions to $ax^2+bx+c=0$?

     $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived from completing the square on the general quadratic equation.

110. Solve for $x$: $x^2-9=0$.

     $x=\pm 3$. Factor as $(x-3)(x+3)=0$.

111. Solve $x^2+2x-8=0$ for $x$.

     $x=2$ or $x=-4$. Factor as $(x+4)(x-2)=0$ or use quadratic formula.

112. Solve for $x$: $(x-3)^2=16$.

     $x=7$ or $x=-1$. $x-3=\pm 4$ gives $x=7$ or $x=-1$.

113. Identify the first step to solve $x^2=49$ over the real numbers.

     Take square roots: $x=\pm\sqrt{49}$. Square root gives both positive and negative values.

114. What are the solutions to $x^4-5x^2=0$?

     $x=0$ or $x=\pm\sqrt{5}$. Factor as $x^2(x^2-5)=0$, giving $x=0$ or $x^2=5$.

115. State the quadratic formula for solutions of $ax^2+bx+c=0$.

     $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general quadratic equation.

116. What is the axis of symmetry formula for $y=ax^2+bx+c$?

     $x=-\frac{b}{2a}$. The parabola's line of symmetry passes through the vertex.

117. What are the real solutions to $x^4-16=0$?

     $x=\pm 2$. Since $x^4=16$, take fourth root: $x=\pm\sqrt[4]{16}=\pm 2$.

118. What substitution can turn $x^4-5x^2+4=0$ into a quadratic equation?

     Let $u=x^2$. Transforms the quartic into $u^2-5u+4=0$ for easier solving.

119. Solve for $x$: $\frac{x+1}{x-2}=3$ and state any restriction on $x$.

     $x=\frac{7}{2}$, with $x\ne 2$. Cross-multiply: $x+1=3(x-2)$, then solve.

120. What are the solutions to $\sqrt{x}=5$?

     $x=25$. Square both sides: $x=25$ (check: $\sqrt{25}=5$ ✓).

121. Solve for $x$: $x^2= -16$ over the real numbers.

     No real solution. Square roots of negative numbers aren't real.

122. Identify the extraneous-solution risk when solving an equation by squaring both sides.

     Squaring can introduce extraneous solutions; check in the original equation. Squaring both sides can create false solutions not in the original.

123. What is the discriminant of $ax^2+bx+c=0$, used to classify the number of real solutions?

     $\Delta=b^2-4ac$. Appears under the square root in the quadratic formula.

124. What does $b^2-4ac>0$ tell you about solutions of $ax^2+bx+c=0$?

     $2$ distinct real solutions. Positive discriminant means the square root yields two different values.

125. What is the axis of symmetry for $f(x)=ax^2+bx+c$?

     $x=-\frac{b}{2a}$. The vertical line through the vertex, found from $-\frac{b}{2a}$.

126. Solve for $x$: $x^2+2x-3=0$.

     $x=1$ or $x=-3$. Factor as $(x-1)(x+3)=0$ or use the quadratic formula.

127. Solve for $x$: $2x^2-8x=0$.

     $x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, so $x=0$ or $x=4$.

128. What does $b^2-4ac<0$ imply about the solutions of $ax^2+bx+c=0$?

     No real solutions (two complex solutions). Negative discriminant means parabola doesn't cross x-axis.

129. What does $\Delta>0$ imply about the real solutions of $ax^2+bx+c=0$?

     $2$ distinct real solutions. Positive discriminant means the square root yields two different values.

130. Solve for $x$: $2x^2-8x=0$.

     $x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, so $x=0$ or $x=4$.

131. Solve for $x$: $x^2-6x+9=0$.

     $x=3$. Perfect square: $(x-3)^2=0$.

132. Find the real solutions of $x^2=49$.

     $x=\pm 7$. $\sqrt{49}=7$, so $x=\pm 7$.

133. What does $b^2-4ac=0$ imply about the real solutions of $ax^2+bx+c=0$?

     One real solution (double root). Zero discriminant means parabola touches x-axis once.

134. What is the discriminant of $ax^2+bx+c=0$?

     $b^2-4ac$. This expression determines the nature and number of real solutions.

135. What substitution is used to solve a quadratic in $x^2$, such as $x^4+5x^2+4=0$?

     Let $u=x^2$. Transforms to $u^2+5u+4=0$, a standard quadratic.

136. What are the solutions to $x^2-6x+9=0$?

     $x=3$. Perfect square $(x-3)^2=0$ has one repeated root.

137. What does $\Delta>0$ tell you about the real solutions of $ax^2+bx+c=0$?

     $2$ distinct real solutions. Positive discriminant means the square root yields two different values.

138. What is the discriminant of $ax^2+bx+c=0$ used to classify solutions?

     $\Delta=b^2-4ac$. Determines the nature of roots; appears under the square root in the quadratic formula.

139. What substitution can turn $ax^4+bx^2+c=0$ into a quadratic?

     Let $u=x^2$. This reduces the quartic to $au^2+bu+c=0$, a quadratic in $u$.

140. What is the solution to the system $y=x^2$ and $y=x+2$?

     $(x,y)=(2,4)$ and $(x,y)=(-1,1)$. Set $x^2=x+2$: $x^2-x-2=0$ factors as $(x-2)(x+1)=0$.

141. Solve for $x$: $x^2=2x+8$.

     $x=4$ or $x=-2$. Rearrange to $x^2-2x-8=0$, factor as $(x-4)(x+2)=0$.

142. What are the solutions of $x^2-9=0$?

     $x=\pm 3$. Factor as $(x-3)(x+3)=0$ and apply zero-product property.

143. Identify the solutions of $(x-5)(x+1)=0$.

     $x=5$ or $x=-1$. Set each factor to zero: $x-5=0$ gives $x=5$, $x+1=0$ gives $x=-1$.

144. What is the zero-product property used when factoring gives $(A)(B)=0$?

     $A=0$ or $B=0$. A product equals zero only when at least one factor equals zero.

145. What are the solutions to $x^2+6x+9=0$?

     $x=-3$. Perfect square $(x+3)^2=0$ has one repeated root.

146. What is the zero-product property used after factoring an equation?

     If $ab=0$, then $a=0$ or $b=0$. At least one factor must equal zero for product to be zero.

147. What does $b^2-4ac>0$ imply about the real solutions of $ax^2+bx+c=0$?

     Two distinct real solutions. Positive discriminant means parabola crosses x-axis twice.

148. Identify the solutions of $\frac{1}{x-2}=3$.

     $x=\frac{7}{3}$. Cross-multiply: $1=3(x-2)$, so $1=3x-6$ and $x=\frac{7}{3}$.

149. What substitution turns $ax^4+bx^2+c=0$ into a quadratic equation?

     Let $u=x^2$. Reduces quartic to quadratic form $au^2+bu+c=0$.

150. What is the solution set of $x^2-5x+6=0$?

     $x=2$ or $x=3$. Factors to $(x-2)(x-3)=0$, then use zero-product property.

151. Identify the $x$-intercepts of a graph in terms of solving an equation.

     Solutions to $f(x)=0$. Where the graph crosses the x-axis, so $y=0$.

152. Solve for $x$: $x^2-5x=0$.

     $x=0$ or $x=5$. Factor out $x$: $x(x-5)=0$.

153. What is the correct first step to solve a rational equation like $\frac{p(x)}{q(x)}=r$?

     Multiply both sides by $q(x)$, with $q(x)\ne 0$. Clears the denominator to create a polynomial equation.

154. What is the key restriction when squaring both sides to solve a radical equation?

     Check for extraneous solutions in the original equation. Squaring can introduce false solutions that don't satisfy the original.

155. What are the solutions to $x^2+4x+3=0$?

     $x=-1$ or $x=-3$. Factor as $(x+1)(x+3)=0$ or use quadratic formula.

156. What is the discriminant for $ax^2+bx+c=0$, used to classify solutions?

     $\Delta=b^2-4ac$. Appears under the square root in the quadratic formula; determines solution types.

157. Identify the substitution that turns $ax^4+bx^2+c=0$ into a quadratic.

     Let $u=x^2$. Reduces the quartic to $au^2+bu+c=0$, a quadratic in $u$.

158. What is the vertex $x$-coordinate of $y=a(x-h)^2+k$?

     $h$. In vertex form, $(h,k)$ is the vertex point.

159. Identify the required restriction when solving $\frac{p(x)}{q(x)}=r$.

     Exclude values where $q(x)=0$. Division by zero is undefined, so check denominator.

160. Solve for $x$: $2x^2-8x=0$.

     $x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, so $x=0$ or $x=4$.

161. Solve for $x$: $2x^2-8=0$.

     $x=\pm 2$. $x^2=4$, so $x=\pm 2$.

162. Solve for $x$: $(x-3)(x+5)=0$.

     $x=3$ or $x=-5$. Apply zero product property: $x-3=0$ or $x+5=0$.

163. What is the solution of $\frac{1}{x}=3$?

     $x=\frac{1}{3}$. Multiply both sides by $x$: $1=3x$, so $x=\frac{1}{3}$.

164. Solve $2^x=8$ for $x$.

     $x=3$. Since $8=2^3$, we have $2^x=2^3$, so $x=3$.

165. What are the solutions to $x^2-9=0$?

     $x=\pm 3$. Factor as $(x-3)(x+3)=0$, then apply zero-product property.

166. Identify the extraneous-solution check after squaring both sides of an equation.

     Substitute solutions into the original equation. Squaring can introduce false solutions that don't satisfy the original.

167. Identify the solutions of $x^2-9=0$.

     $x=\pm 3$. Factor as $(x-3)(x+3)=0$, then apply zero-product property.

168. What is the domain restriction when solving $\sqrt{g(x)}=c$ over the reals?

     $g(x)\ge 0$. Square roots require non-negative values under the radical.

169. What does it mean if $\Delta=0$ for $ax^2+bx+c=0$?

     $1$ real solution (double root). Zero discriminant means the parabola touches the x-axis once.

170. What is the solution of $2x^2-8x=0$?

     $x=0$ and $x=4$. Factor out $2x$: $2x(x-4)=0$, giving $x=0$ or $x=4$.

171. What is the zero-product property used when solving factored equations?

     If $ab=0$, then $a=0$ or $b=0$. At least one factor must be zero for product to be zero.

172. What is the vertex $x$-coordinate of $y=ax^2+bx+c$?

     $x=-\frac{b}{2a}$. The axis of symmetry passes through the vertex.

173. What are the solutions to $2x^2-8x=0$?

     $x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, so $x=0$ or $x=4$.

174. What does it mean if $\Delta>0$ for $ax^2+bx+c=0$?

     $2$ distinct real solutions. Positive discriminant means the parabola crosses the x-axis twice.

175. What is the solution of $x^2-9=0$?

     $x=\pm 3$. Factor as $(x-3)(x+3)=0$, then apply zero-product property.

176. What is the standard form of a quadratic function?

     $f(x)=ax^2+bx+c$. The general form with coefficients $a$, $b$, and $c$.

177. What is the key first step to solve an equation containing $\sqrt{\,}$ terms?

     Isolate the radical term before squaring. Ensures only the radical is on one side before eliminating it.

178. What is the vertex $x$-coordinate formula for $y=ax^2+bx+c$?

     $x=-\frac{b}{2a}$. The axis of symmetry passes through the vertex of the parabola.

179. What does $\Delta=0$ tell you about the real solutions of $ax^2+bx+c=0$?

     $1$ real double root. Zero discriminant makes $\pm\sqrt{0}=0$, giving one repeated solution.

180. What is the zero-product property used to solve $(x-p)(x-q)=0$?

     $x=p$ or $x=q$. If a product equals zero, at least one factor must be zero.

181. Identify the method to solve $a^x=b$ for $x$ when $a>0$, $a\ne^1$, $b>0$.

     Take logs: $x=\frac{\log(b)}{\log(a)}$. Logarithms convert exponential equations to linear form.

182. State the vertex form of a quadratic function.

     $y=a(x-h)^2+k$. Shows the vertex at $(h,k)$ and vertical stretch factor $a$.

183. What is the equation of a parabola with vertex $(h,k)$ opening up or down?

     $y=a(x-h)^2+k$. The vertex form shows transformations from $y=x^2$.

184. Identify the solutions of $x^2+6x+9=0$.

     $x=-3$. Perfect square $(x+3)^2=0$ has one repeated root.

185. What are the solutions to $x^4-5x^2+4=0$ over the real numbers?

     $x=\pm 1$ or $x=\pm 2$. Let $u=x^2$: $u^2-5u+4=0$ gives $u=1,4$, so $x=\pm 1, \pm 2$.

186. What is the solution of $2x^2-8x=0$?

     $x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, so $x=0$ or $x=4$.

187. Solve $\sqrt{x+5}=4$ for $x$.

     $x=11$. Square both sides: $x+5=16$, so $x=11$.

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