Solve Quadratics by Multiple Methods - Algebra 1

1. How do you rewrite $\sqrt{-k}$ for real $k>0$ using $i$?

   $\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the square root.

2. Identify the solution type from $\Delta=0$.

   One real double solution. Zero discriminant means one repeated solution.

3. What is the discriminant of $x^2-6x+9=0$?

   $0$. $b^2-4ac=36-36=0$ for perfect square.

4. Identify the solution type from $\Delta=25$.

   Two distinct real solutions. Positive discriminant means two real solutions.

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

   $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard formula derived from completing the square on $ax^2+bx+c=0$.

6. What are the solutions of $x^2=0$?

   $x=0$. Only zero squared equals zero.

7. What number do you add and subtract to complete the square for $x^2+bx$?

   $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.

8. What are the solutions of $(x-r)(x-s)=0$?

   $x=r$ or $x=s$. Set each factor equal to zero and solve.

9. What is the imaginary unit $i$ defined as?

   $i=\sqrt{-1}$. Fundamental definition of the imaginary unit.

10. Solve using the quadratic formula: $2x^2+4x+1=0$.

    $x=-1\pm\frac{\sqrt{2}}{2}$. Apply formula with $a=2$, $b=4$, $c=1$.

11. Solve by factoring: $4x^2-25=0$.

    $x=-\frac{5}{2}$ or $x=\frac{5}{2}$. Difference of squares: $(2x-5)(2x+5)=0$.

12. What does $\Delta>0$ tell you about the solutions of a quadratic?

    Two distinct real solutions. Positive discriminant means two real roots exist.

13. Solve by factoring: $x^2-4x+4=0$.

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

14. Solve by factoring: $x^2-7x+12=0$.

    $x=3$ or $x=4$. Find factors of 12 that add to $-7$: $(x-3)(x-4)=0$.

15. Solve using the quadratic formula: $x^2-2x-3=0$.

    $x=-1$ or $x=3$. Apply formula with $a=1$, $b=-2$, $c=-3$.

16. Solve by completing the square: $x^2-4x-1=0$.

    $x=2\pm\sqrt{5}$. Complete square: $(x-2)^2=5$, so $x=2\pm\sqrt{5}$.

17. What is the factored form pattern for a monic quadratic $x^2+bx+c$?

    $(x+m)(x+n)$ with $m+n=b$ and $mn=c$. Two numbers that add to $b$ and multiply to $c$.

18. What property allows you to set each factor to $0$ in $(x-r)(x-s)=0$?

    Zero Product Property. If a product equals zero, at least one factor must be zero.

19. What is $\sqrt{\frac{49}{16}}$?

    $\frac{7}{4}$. Square root of fraction: $\sqrt{\frac{49}{16}}=\frac{7}{4}$.

20. What is $\sqrt{36}$?

    $6$. Perfect square: $36=6^2$.

21. What is $\sqrt{-81}$ written using $i$?

    $9i$. $\sqrt{-81}=\sqrt{81}\cdot i=9i$.

22. What does $\Delta<0$ tell you about the solutions of a quadratic?

    Two complex (nonreal) solutions. Negative discriminant means roots involve imaginary numbers.

23. What is the key first step to solve $ax^2=k$ by taking square roots?

    Divide by $a$ to get $x^2=\frac{k}{a}$. Isolate $x^2$ to apply square root method.

24. What is the vertex form used after completing the square?

    $a(x-h)^2+k=0$ or $a(x-h)^2=-k$. Vertex form after completing the square process.

25. What are the solutions of $x^2=-m$ for real $m>0$?

    $x=\pm i\sqrt{m}$. Negative under square root creates imaginary solutions.

26. Solve by factoring: $3x^2-12x=0$.

    $x=0$ or $x=4$. Factor out $3x$: $3x(x-4)=0$.

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

    $x=\pm\frac{1}{2}$. Perfect square: $\frac{1}{4}=\left(\frac{1}{2}\right)^2$.

28. What is the discriminant of $x^2+2x+10=0$?

    $-36$. $b^2-4ac=4-40=-36$ gives complex solutions.

29. What is the square root of $a^2$ for real $a$?

    $\sqrt{a^2}=|a|$. Principal square root is always non-negative.

30. Solve by factoring: $2x^2+7x+3=0$.

    $x=-3$ or $x=-\frac{1}{2}$. Factor as $(2x+1)(x+3)=0$.

31. Solve by factoring: $x^2+5x+6=0$.

    $x=-3$ or $x=-2$. Find factors of 6 that add to 5: $(x+3)(x+2)=0$.

32. Solve by taking square roots: $(x-5)^2=0$.

    $x=5$. Perfect square equals zero has one solution.

33. What is the discriminant for $ax^2+bx+c=0$?

    $\Delta=b^2-4ac$. Expression under the square root that determines solution types.

34. Identify the solution type from $\Delta=-9$.

    Two complex (nonreal) solutions. Negative discriminant means complex solutions.

35. What does $\Delta=0$ tell you about the solutions of a quadratic?

    One real double solution. Zero discriminant means the quadratic has one repeated root.

36. Solve by factoring: $x^2+2x-15=0$.

    $x=-5$ or $x=3$. Find factors of $-15$ that add to 2: $(x+5)(x-3)=0$.

37. Solve by factoring: $x^2-9=0$.

    $x=-3$ or $x=3$. Difference of squares: $(x-3)(x+3)=0$.

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