Question $5 c^{2}+4 c=6$

Studdy Solution

STEP 1

1. The equation $5c^2 + 4c = 6$ is a quadratic equation.
2. The standard form of a quadratic equation is $ax^2 + bx + c = 0$ .
3. Solving the quadratic equation will involve rearranging it into standard form and using the quadratic formula or factoring.

STEP 2

1. Rearrange the equation into standard form.
2. Solve the quadratic equation using the quadratic formula.
3. Simplify the solutions.

STEP 3

Rearrange the given equation into the standard form of a quadratic equation by moving all terms to one side:
$5c^2 + 4c - 6 = 0$

STEP 4

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $ax^2 + bx + c = 0$ :
$a = 5, \quad b = 4, \quad c = -6$

STEP 5

Use the quadratic formula to solve for $c$ . The quadratic formula is given by:
$c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Substitute the values of $a$ , $b$ , and $c$ into the formula:
$c = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 5 \cdot (-6)}}{2 \cdot 5}$

STEP 6

Calculate the discriminant $b^2 - 4ac$ :
$b^2 - 4ac = 4^2 - 4 \cdot 5 \cdot (-6)$ $= 16 + 120$ $= 136$

STEP 7

Substitute the discriminant back into the quadratic formula and simplify:
$c = \frac{-4 \pm \sqrt{136}}{10}$

STEP 8

Simplify the square root of the discriminant:
$\sqrt{136} = \sqrt{4 \times 34} = 2\sqrt{34}$
Substitute back into the equation:
$c = \frac{-4 \pm 2\sqrt{34}}{10}$

STEP 9

Simplify the expression by factoring out the common factor in the numerator:
$c = \frac{-2 \pm \sqrt{34}}{5}$
The solutions for $c$ are:
$c = \frac{-2 + \sqrt{34}}{5} \quad \text{and} \quad c = \frac{-2 - \sqrt{34}}{5}$

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