Question Solve the quadratic equation $-7x^2 - 2x + 6 = 0$ and find its roots.

Studdy Solution

STEP 1

Assumptions
1. We have a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. The given quadratic equation is $-7x^2 - 2x + 6 = 0$ .
3. We will use the quadratic formula to find the roots of the equation, which is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

STEP 2

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $-7x^2 - 2x + 6 = 0$ .
$a = -7, \quad b = -2, \quad c = 6$

STEP 3

Calculate the discriminant using the formula $b^2 - 4ac$ .
$\Delta = b^2 - 4ac$

STEP 4

Substitute the values of $a$ , $b$ , and $c$ into the discriminant formula.
$\Delta = (-2)^2 - 4(-7)(6)$

STEP 5

Calculate the value of the discriminant.
$\Delta = 4 + 168 = 172$

STEP 6

Since the discriminant $\Delta > 0$ , we know that there are two distinct real roots for the quadratic equation.

STEP 7

Now, apply the quadratic formula to find the two roots.
$x = \frac{-b \pm \sqrt{\Delta}}{2a}$

STEP 8

Substitute the values of $a$ , $b$ , and the discriminant $\Delta$ into the quadratic formula.
$x = \frac{-(-2) \pm \sqrt{172}}{2(-7)}$

STEP 9

Simplify the expression inside the quadratic formula.
$x = \frac{2 \pm \sqrt{172}}{-14}$

STEP 10

Since $\sqrt{172}$ can be simplified to $2\sqrt{43}$ , we rewrite the expression.
$x = \frac{2 \pm 2\sqrt{43}}{-14}$

STEP 11

Now, we can simplify the fraction by dividing the numerator and the denominator by 2.
$x = \frac{1 \pm \sqrt{43}}{-7}$

STEP 12

Finally, we write down the two roots separately.
Root 1: $x_1 = \frac{1 + \sqrt{43}}{-7}$
Root 2: $x_2 = \frac{1 - \sqrt{43}}{-7}$
These are the two roots of the quadratic equation $-7x^2 - 2x + 6 = 0$ .

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