Question Find the exact solutions for the quadratic equation $x^{2} + 8x = 6$ .

Studdy Solution

STEP 1

1. The equation $x^{2}+8x=6$ is a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. The roots of the quadratic equation can be found using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
3. The discriminant $D = b^2 - 4ac$ will determine the nature of the roots (real and distinct, real and equal, or complex).

STEP 2

1. Rewrite the equation in standard quadratic form.
2. Calculate the discriminant.
3. Use the quadratic formula to find the roots.
4. Simplify the roots to their exact form.

STEP 3

Rewrite the given equation $x^{2}+8x=6$ in standard quadratic form by subtracting 6 from both sides to get $x^{2}+8x-6=0$ .

STEP 4

Calculate the discriminant $D$ using the formula $D = b^2 - 4ac$ , where $a=1$ , $b=8$ , and $c=-6$ .

STEP 5

Substitute the values of $a$ , $b$ , and $c$ into the discriminant formula to get $D = 8^2 - 4(1)(-6)$ .

STEP 6

Simplify the discriminant to get $D = 64 + 24$ .

STEP 7

Further simplify the discriminant to get $D = 88$ .

STEP 8

Use the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$ to find the roots, with $a=1$ , $b=8$ , and $D=88$ .

STEP 9

Substitute the values into the quadratic formula to get $x = \frac{-8 \pm \sqrt{88}}{2}$ .

STEP 10

Simplify the square root $\sqrt{88}$ by factoring it into $\sqrt{4 \cdot 22}$ .

STEP 11

Further simplify the square root to get $\sqrt{88} = \sqrt{4} \cdot \sqrt{22} = 2\sqrt{22}$ .

STEP 12

Substitute the simplified square root back into the formula to get $x = \frac{-8 \pm 2\sqrt{22}}{2}$ .

STEP 13

Simplify the expression by dividing all terms by 2 to get $x = -4 \pm \sqrt{22}$ .

STEP 14

Write the final exact roots as two separate numbers: $x_1 = -4 + \sqrt{22}$ and $x_2 = -4 - \sqrt{22}$ .
The exact roots of the equation $x^{2}+8x=6$ are: $x_1 = -4 + \sqrt{22}, \quad x_2 = -4 - \sqrt{22}$

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