QuestionSolve by using the quadratic formula: $6 x^{2}+3 x-7=0$ .

Studdy Solution

STEP 1

1. The equation $6x^2 + 3x - 7 = 0$ is a quadratic equation.
2. The quadratic formula can be used to find the roots of a quadratic equation of the form $ax^2 + bx + c = 0$ .
3. The coefficients for this equation are $a = 6$ , $b = 3$ , and $c = -7$ .

STEP 2

1. Identify the coefficients $a$ , $b$ , and $c$ .
2. Write down the quadratic formula.
3. Substitute the coefficients into the quadratic formula.
4. Simplify the expression under the square root (the discriminant).
5. Calculate the roots using the quadratic formula.

STEP 3

Identify the coefficients from the quadratic equation $6x^2 + 3x - 7 = 0$ :
$a = 6, \quad b = 3, \quad c = -7$

STEP 4

Write down the quadratic formula, which is used to find the roots of a quadratic equation $ax^2 + bx + c = 0$ :
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

STEP 5

Substitute the identified coefficients $a = 6$ , $b = 3$ , and $c = -7$ into the quadratic formula:
$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 6 \cdot (-7)}}{2 \cdot 6}$

STEP 6

Simplify the expression under the square root, known as the discriminant:
$b^2 - 4ac = 3^2 - 4 \cdot 6 \cdot (-7)$ $= 9 + 168$ $= 177$

STEP 7

Calculate the roots using the quadratic formula with the simplified discriminant:
$x = \frac{-3 \pm \sqrt{177}}{12}$
This gives us two solutions:
$x_1 = \frac{-3 + \sqrt{177}}{12}$ $x_2 = \frac{-3 - \sqrt{177}}{12}$
The solutions to the quadratic equation are:
$x_1 = \frac{-3 + \sqrt{177}}{12}$ $x_2 = \frac{-3 - \sqrt{177}}{12}$

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