Question

1. $x^2 + x - 56 = 0$
2. $3x^2 - 8x = x^2 + 42$
3. $9p^2 - 1 = 0$
4. $2w^2 + 29 = -21$
5. $x^2 + 4x + 8 = 0$
6. $n^2 + 15n + 43 = 0$

Explanation:

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Problem 1: $x^2 + x - 56 = 0$

Step1: Factor the quadratic

Find two numbers that multiply to $-56$ and add to $1$: $8$ and $-7$.
$(x + 8)(x - 7) = 0$

Step2: Solve for $x$

Set each factor equal to 0:
$x + 8 = 0 \implies x = -8$
$x - 7 = 0 \implies x = 7$

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Problem 3: $3x^2 - 8x = x^2 + 42$

Step1: Simplify to standard form

Subtract $x^2$ and $42$ from both sides:
$3x^2 - x^2 - 8x - 42 = 0$
$2x^2 - 8x - 42 = 0$
Divide by 2: $x^2 - 4x - 21 = 0$

Step2: Factor the quadratic

Find two numbers that multiply to $-21$ and add to $-4$: $-7$ and $3$.
$(x - 7)(x + 3) = 0$

Step3: Solve for $x$

$x - 7 = 0 \implies x = 7$
$x + 3 = 0 \implies x = -3$

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Problem 5: $9p^2 - 1 = 0$

Step1: Rewrite as difference of squares

$9p^2 = (3p)^2$, $1 = 1^2$, so:
$(3p - 1)(3p + 1) = 0$

Step2: Solve for $p$

$3p - 1 = 0 \implies p = \frac{1}{3}$
$3p + 1 = 0 \implies p = -\frac{1}{3}$

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Problem 7: $2w^2 + 29 = -21$

Step1: Isolate the squared term

Subtract 29 from both sides:
$2w^2 = -21 - 29$
$2w^2 = -50$
Divide by 2: $w^2 = -25$

Step2: Solve for $w$

Take square roots of both sides (using imaginary unit $i = \sqrt{-1}$):
$w = \pm\sqrt{-25} = \pm 5i$

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Problem 9: $x^2 + 4x + 8 = 0$

Step1: Use quadratic formula

For $ax^2+bx+c=0$, $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$. Here $a=1, b=4, c=8$.
$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(8)}}{2(1)}$

Step2: Calculate discriminant and solve

$\sqrt{16 - 32} = \sqrt{-16} = 4i$
$x = \frac{-4 \pm 4i}{2} = -2 \pm 2i$

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Problem 11: $n^2 + 15n + 93 = 0$

Step1: Use quadratic formula

Here $a=1, b=15, c=93$.
$n = \frac{-15 \pm \sqrt{15^2 - 4(1)(93)}}{2(1)}$

Step2: Calculate discriminant and solve

$\sqrt{225 - 372} = \sqrt{-147} = \sqrt{49 \times (-3)} = 7i\sqrt{3}$
$n = \frac{-15 \pm 7i\sqrt{3}}{2}$

Answer:

1. $x = -8$ or $x = 7$
2. $x = 7$ or $x = -3$
3. $p = \frac{1}{3}$ or $p = -\frac{1}{3}$
4. $w = 5i$ or $w = -5i$
5. $x = -2 + 2i$ or $x = -2 - 2i$
6. $n = \frac{-15 + 7i\sqrt{3}}{2}$ or $n = \frac{-15 - 7i\sqrt{3}}{2}$