Question

name:
quadratic formula
determine the solution for each quadratic equation. find your answer in the bank and color the corresponding number on the picture.
solve for x in simplest form
$x^2 - 5x - 15 = 0$
①
solve for x in simplest form
$3x^2 - 3x - 6 = -2$
②
solve for x in simplest form
$4x^2 + 2x - 2 = 3x$
③
solve for x in simplest form
$5x^2 + 12x + 6 = 0$
④
solve for x in simplest form
$3x^2 - 6x + 2 = 3$
⑤
solve for x in simplest form
$6x^2 + 6x - 7 = 4x^2$
⑥
solve for x in simplest form
$x^2 + 2x - 20 = -3$
⑦
solve for x in simplest form
$6x^2 + 10x + 2 = x^2$
⑧
solve for x in simplest form
$x^2 + 2x - 11 = 0$
⑨
solve for x in simplest form
$x^2 + 3x - 7 = x$
⑩
solve for x in simplest form
$3x^2 + 2x - 1 = -2x$
⑪
solve for x in simplest form
$5x^2 - 18x + 12 = 0$
⑫

Explanation:

Quadratic Formula Reference

For a quadratic equation $ax^2 + bx + c = 0$, the solutions are:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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Problem 1: $x^2 - 5x - 15 = 0$

Step1: Identify a, b, c

$a=1$, $b=-5$, $c=-15$

Step2: Compute discriminant

$\Delta = (-5)^2 - 4(1)(-15) = 25 + 60 = 85$

Step3: Apply quadratic formula

$x = \frac{5 \pm \sqrt{85}}{2(1)}$

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Problem 2: $3x^2 - 3x - 6 = -2$

Step1: Rewrite in standard form

$3x^2 - 3x - 4 = 0$

Step2: Identify a, b, c

$a=3$, $b=-3$, $c=-4$

Step3: Compute discriminant

$\Delta = (-3)^2 - 4(3)(-4) = 9 + 48 = 57$

Step4: Apply quadratic formula

$x = \frac{3 \pm \sqrt{57}}{2(3)} = \frac{3 \pm \sqrt{57}}{6}$

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Problem 3: $4x^2 + 2x - 2 = 3x$

Step1: Rewrite in standard form

$4x^2 - x - 2 = 0$

Step2: Identify a, b, c

$a=4$, $b=-1$, $c=-2$

Step3: Compute discriminant

$\Delta = (-1)^2 - 4(4)(-2) = 1 + 32 = 33$

Step4: Apply quadratic formula

$x = \frac{1 \pm \sqrt{33}}{2(4)} = \frac{1 \pm \sqrt{33}}{8}$

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Problem 4: $5x^2 + 12x + 6 = 0$

Step1: Identify a, b, c

$a=5$, $b=12$, $c=6$

Step2: Compute discriminant

$\Delta = (12)^2 - 4(5)(6) = 144 - 120 = 24$

Step3: Simplify discriminant

$\sqrt{24} = 2\sqrt{6}$

Step4: Apply quadratic formula

$x = \frac{-12 \pm 2\sqrt{6}}{2(5)} = \frac{-6 \pm \sqrt{6}}{5}$

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Problem 5: $3x^2 - 6x + 2 = 3$

Step1: Rewrite in standard form

$3x^2 - 6x - 1 = 0$

Step2: Identify a, b, c

$a=3$, $b=-6$, $c=-1$

Step3: Compute discriminant

$\Delta = (-6)^2 - 4(3)(-1) = 36 + 12 = 48$

Step4: Simplify discriminant

$\sqrt{48} = 4\sqrt{3}$

Step5: Apply quadratic formula

$x = \frac{6 \pm 4\sqrt{3}}{2(3)} = \frac{3 \pm 2\sqrt{3}}{3}$

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Problem 6: $6x^2 + 6x - 7 = 4x^2$

Step1: Rewrite in standard form

$2x^2 + 6x - 7 = 0$

Step2: Identify a, b, c

$a=2$, $b=6$, $c=-7$

Step3: Compute discriminant

$\Delta = (6)^2 - 4(2)(-7) = 36 + 56 = 92$

Step4: Simplify discriminant

$\sqrt{92} = 2\sqrt{23}$

Step5: Apply quadratic formula

$x = \frac{-6 \pm 2\sqrt{23}}{2(2)} = \frac{-3 \pm \sqrt{23}}{2}$

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Problem 7: $x^2 + 2x - 20 = -3$

Step1: Rewrite in standard form

$x^2 + 2x - 17 = 0$

Step2: Identify a, b, c

$a=1$, $b=2$, $c=-17$

Step3: Compute discriminant

$\Delta = (2)^2 - 4(1)(-17) = 4 + 68 = 72$

Step4: Simplify discriminant

$\sqrt{72} = 6\sqrt{2}$

Step5: Apply quadratic formula

$x = \frac{-2 \pm 6\sqrt{2}}{2(1)} = -1 \pm 3\sqrt{2}$

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Problem 8: $6x^2 + 10x + 2 = x^2$

Step1: Rewrite in standard form

$5x^2 + 10x + 2 = 0$

Step2: Identify a, b, c

$a=5$, $b=10$, $c=2$

Step3: Compute discriminant

$\Delta = (10)^2 - 4(5)(2) = 100 - 40 = 60$

Step4: Simplify discriminant

$\sqrt{60} = 2\sqrt{15}$

Step5: Apply quadratic formula

$x = \frac{-10 \pm 2\sqrt{15}}{2(5)} = \frac{-5 \pm \sqrt{15}}{5}$

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Problem 9: $x^2 + 2x - 11 = 0$

Step1: Identify a, b, c

$a=1$, $b=2$, $c=-11$

Step2: Compute discriminant

$\Delta = (2)^2 - 4(1)(-11) = 4 + 44 = 48$

Step3: Simplify discriminant

$\sqrt{48} = 4\sqrt{3}$

Step4: Apply quadratic formula

$x = \frac{-2 \pm 4\sqrt{3}}{2(1)} = -1 \pm 2\sqrt{3}$

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Problem 10: $x^2 + 3x - 7 = x$

Step1: Rewrite in standard form

$x^2 + 2x - 7 = 0$

Step2: Identify a, b, c

$a=1$, $b=2$, $c=-7$

Step3: Compute discriminant

$\Delta = (2)^2 - 4(1)(-7) = 4 + 28 = 32$

Step4: Simplify discriminant

$\sqrt{32} = 4\sqrt{2}$

Step5: Apply quadratic formula

$x = \frac{-2 \pm 4\sqrt{2}}{2(1)} = -1 \pm 2\sqrt{2}$

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Problem 11: $3x^2 + 2x - 1 = -2x$

Step1: Rewrite in standard form

$3x^2 + 4x - 1 = 0$

Step2: Identify a, b, c

$a=3$, $b=4$, $c=-1$

Step3: Compute discriminant

$\Delta = (4)^2 - 4(3)(-1) = 16 + 12 = 28$

Step4: Simplify discriminant

$\sqrt{28} = 2\sqrt{7}$

Step5: Apply quadratic formula

$x = \frac{-4 \pm 2\sqrt{7}}{2(3)} = \frac{-2 \pm \sqrt{7}}{3}$

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Problem 12: $5x^2 - 18x + 12 = 0$

Step1: Identify a, b, c

$a=5$, $b=-18$, $c=12$

Step2: Compute discriminant

$\Delta = (-18)^2 - 4(5)(12) = 324 - 240 = 84$

Step3: Simplify discriminant

$\sqrt{84} = 2\sqrt{21}$

Step4: Apply quadratic formula

$x = \frac{18 \pm 2\sqrt{21}}{2(5)} = \frac{9 \pm \sqrt{21}}{5}$

Answer:

1. $x = \frac{5 \pm \sqrt{85}}{2}$
2. $x = \frac{3 \pm \sqrt{57}}{6}$
3. $x = \frac{1 \pm \sqrt{33}}{8}$
4. $x = \frac{-6 \pm \sqrt{6}}{5}$
5. $x = \frac{3 \pm 2\sqrt{3}}{3}$
6. $x = \frac{-3 \pm \sqrt{23}}{2}$
7. $x = -1 \pm 3\sqrt{2}$
8. $x = \frac{-5 \pm \sqrt{15}}{5}$
9. $x = -1 \pm 2\sqrt{3}$
10. $x = -1 \pm 2\sqrt{2}$
11. $x = \frac{-2 \pm \sqrt{7}}{3}$
12. $x = \frac{9 \pm \sqrt{21}}{5}$