Solve Quadratics by Multiple Methods - Algebra 2
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What does $\Delta<0$ guarantee about solutions of $ax^2+bx+c=0$?
What does $\Delta<0$ guarantee about solutions of $ax^2+bx+c=0$?
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Two complex conjugate solutions. Negative discriminant means the parabola doesn't cross the x-axis.
Two complex conjugate solutions. Negative discriminant means the parabola doesn't cross the x-axis.
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Solve by factoring: $x^2-4x+3=0$.
Solve by factoring: $x^2-4x+3=0$.
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$x=1$ or $x=3$. Factor: $(x-1)(x-3)=0$, then use zero-product property.
$x=1$ or $x=3$. Factor: $(x-1)(x-3)=0$, then use zero-product property.
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What is $\Delta$ for $2x^2-3x-2=0$?
What is $\Delta$ for $2x^2-3x-2=0$?
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$\Delta=25$. $\Delta=(-3)^2-4(2)(-2)=9+16=25$.
$\Delta=25$. $\Delta=(-3)^2-4(2)(-2)=9+16=25$.
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Solve using the quadratic formula: $2x^2+4x+5=0$.
Solve using the quadratic formula: $2x^2+4x+5=0$.
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$x=-1\pm\frac{\sqrt{6}}{2}i$. Use $a=2$, $b=4$, $c=5$; $\sqrt{-24}=2\sqrt{6}i$.
$x=-1\pm\frac{\sqrt{6}}{2}i$. Use $a=2$, $b=4$, $c=5$; $\sqrt{-24}=2\sqrt{6}i$.
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Solve using the quadratic formula: $x^2+2x+5=0$.
Solve using the quadratic formula: $x^2+2x+5=0$.
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$x=-1\pm 2i$. Negative discriminant gives complex solutions: $\sqrt{-16}=4i$.
$x=-1\pm 2i$. Negative discriminant gives complex solutions: $\sqrt{-16}=4i$.
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Solve using the quadratic formula: $x^2-2x+5=0$.
Solve using the quadratic formula: $x^2-2x+5=0$.
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$x=1\pm 2i$. Negative discriminant gives complex solutions: $\sqrt{-16}=4i$.
$x=1\pm 2i$. Negative discriminant gives complex solutions: $\sqrt{-16}=4i$.
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Solve using the quadratic formula: $3x^2-12x+12=0$.
Solve using the quadratic formula: $3x^2-12x+12=0$.
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$x=2$. Use $a=3$, $b=-12$, $c=12$; discriminant equals 0.
$x=2$. Use $a=3$, $b=-12$, $c=12$; discriminant equals 0.
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Solve using the quadratic formula: $2x^2+3x-2=0$.
Solve using the quadratic formula: $2x^2+3x-2=0$.
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$x=\frac{1}{2}$ or $x=-2$. Use $a=2$, $b=3$, $c=-2$ in the quadratic formula.
$x=\frac{1}{2}$ or $x=-2$. Use $a=2$, $b=3$, $c=-2$ in the quadratic formula.
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Solve using the quadratic formula: $x^2+4x+1=0$.
Solve using the quadratic formula: $x^2+4x+1=0$.
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$x=-2\pm\sqrt{3}$. Use $a=1$, $b=4$, $c=1$ in the quadratic formula.
$x=-2\pm\sqrt{3}$. Use $a=1$, $b=4$, $c=1$ in the quadratic formula.
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Solve using the quadratic formula: $x^2+5x+6=0$.
Solve using the quadratic formula: $x^2+5x+6=0$.
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$x=-2$ or $x=-3$. Use $a=1$, $b=5$, $c=6$ in the quadratic formula.
$x=-2$ or $x=-3$. Use $a=1$, $b=5$, $c=6$ in the quadratic formula.
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Solve by completing the square: $x^2+2x-8=0$.
Solve by completing the square: $x^2+2x-8=0$.
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$x=2$ or $x=-4$. Complete the square: $(x+1)^2=9$, so $x+1=\pm 3$.
$x=2$ or $x=-4$. Complete the square: $(x+1)^2=9$, so $x+1=\pm 3$.
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State the quadratic formula for solutions of $ax^2+bx+c=0$.
State the quadratic formula for solutions of $ax^2+bx+c=0$.
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$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard form for finding roots of any quadratic equation.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard form for finding roots of any quadratic equation.
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What is the discriminant for $ax^2+bx+c=0$?
What is the discriminant for $ax^2+bx+c=0$?
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$\Delta=b^2-4ac$. The discriminant determines the nature of quadratic solutions.
$\Delta=b^2-4ac$. The discriminant determines the nature of quadratic solutions.
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Solve by completing the square: $x^2-4x-1=0$.
Solve by completing the square: $x^2-4x-1=0$.
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$x=2\pm\sqrt{5}$. Complete the square: $(x-2)^2=5$, so $x-2=\pm\sqrt{5}$.
$x=2\pm\sqrt{5}$. Complete the square: $(x-2)^2=5$, so $x-2=\pm\sqrt{5}$.
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What does $\Delta>0$ guarantee about solutions of $ax^2+bx+c=0$?
What does $\Delta>0$ guarantee about solutions of $ax^2+bx+c=0$?
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Two distinct real solutions. Positive discriminant means the parabola crosses the x-axis twice.
Two distinct real solutions. Positive discriminant means the parabola crosses the x-axis twice.
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What does $\Delta=0$ guarantee about solutions of $ax^2+bx+c=0$?
What does $\Delta=0$ guarantee about solutions of $ax^2+bx+c=0$?
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One real double solution. Zero discriminant means the parabola touches the x-axis at one point.
One real double solution. Zero discriminant means the parabola touches the x-axis at one point.
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What does $\Delta<0$ guarantee about solutions of $ax^2+bx+c=0$?
What does $\Delta<0$ guarantee about solutions of $ax^2+bx+c=0$?
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Two complex conjugate solutions. Negative discriminant means the parabola doesn't cross the x-axis.
Two complex conjugate solutions. Negative discriminant means the parabola doesn't cross the x-axis.
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State the square root property used to solve $(x-h)^2=k$.
State the square root property used to solve $(x-h)^2=k$.
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$(x-h)^2=k\Rightarrow x-h=\pm\sqrt{k}$. Take the square root of both sides to isolate $x-h$.
$(x-h)^2=k\Rightarrow x-h=\pm\sqrt{k}$. Take the square root of both sides to isolate $x-h$.
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What are the solutions of $x^2=49$?
What are the solutions of $x^2=49$?
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$x=\pm 7$. Take the square root of both sides: $\sqrt{49}=7$.
$x=\pm 7$. Take the square root of both sides: $\sqrt{49}=7$.
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What are the solutions of $(x-3)^2=16$?
What are the solutions of $(x-3)^2=16$?
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$x=7$ or $x=-1$. Square root both sides: $x-3=\pm 4$, so $x=3\pm 4$.
$x=7$ or $x=-1$. Square root both sides: $x-3=\pm 4$, so $x=3\pm 4$.
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What are the solutions of $(x+5)^2=9$?
What are the solutions of $(x+5)^2=9$?
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$x=-2$ or $x=-8$. Square root both sides: $x+5=\pm 3$, so $x=-5\pm 3$.
$x=-2$ or $x=-8$. Square root both sides: $x+5=\pm 3$, so $x=-5\pm 3$.
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What are the solutions of $2(x-1)^2=18$?
What are the solutions of $2(x-1)^2=18$?
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$x=4$ or $x=-2$. Divide by 2 first: $(x-1)^2=9$, then $x-1=\pm 3$.
$x=4$ or $x=-2$. Divide by 2 first: $(x-1)^2=9$, then $x-1=\pm 3$.
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What are the solutions of $(x-4)^2=0$?
What are the solutions of $(x-4)^2=0$?
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$x=4$. Only one solution since the square root of 0 is 0.
$x=4$. Only one solution since the square root of 0 is 0.
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What are the solutions of $x^2+8x+16=0$?
What are the solutions of $x^2+8x+16=0$?
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$x=-4$. Perfect square trinomial: $(x+4)^2=0$, so $x=-4$.
$x=-4$. Perfect square trinomial: $(x+4)^2=0$, so $x=-4$.
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Factor and solve $x^2-9=0$.
Factor and solve $x^2-9=0$.
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$x=\pm 3$. Difference of squares: $(x-3)(x+3)=0$.
$x=\pm 3$. Difference of squares: $(x-3)(x+3)=0$.
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Factor and solve $x^2-5x=0$.
Factor and solve $x^2-5x=0$.
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$x=0$ or $x=5$. Factor out $x$: $x(x-5)=0$, then use zero-product property.
$x=0$ or $x=5$. Factor out $x$: $x(x-5)=0$, then use zero-product property.
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Factor and solve $x^2+7x+12=0$.
Factor and solve $x^2+7x+12=0$.
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$x=-3$ or $x=-4$. Factor: $(x+3)(x+4)=0$, then use zero-product property.
$x=-3$ or $x=-4$. Factor: $(x+3)(x+4)=0$, then use zero-product property.
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Factor and solve $x^2-6x+8=0$.
Factor and solve $x^2-6x+8=0$.
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$x=2$ or $x=4$. Factor: $(x-2)(x-4)=0$, then use zero-product property.
$x=2$ or $x=4$. Factor: $(x-2)(x-4)=0$, then use zero-product property.
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Factor and solve $x^2+x-6=0$.
Factor and solve $x^2+x-6=0$.
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$x=2$ or $x=-3$. Factor: $(x+3)(x-2)=0$, then use zero-product property.
$x=2$ or $x=-3$. Factor: $(x+3)(x-2)=0$, then use zero-product property.
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Solve by factoring: $2x^2-8x=0$.
Solve by factoring: $2x^2-8x=0$.
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$x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, then use zero-product property.
$x=0$ or $x=4$. Factor out $2x$: $2x(x-4)=0$, then use zero-product property.
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