Question Solve for $w$ in the quadratic equation $5 w^{2} = 13 w + 6$ .

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic equation in the form of $5w^2 = 13w + 6$ .
2. We need to solve for the variable $w$ .

STEP 2

To solve the quadratic equation, we first need to set it to the standard form $ax^2 + bx + c = 0$ . We can do this by bringing all terms to one side of the equation.
$5w^2 - 13w - 6 = 0$

STEP 3

Now, we need to factor the quadratic equation if possible, or use the quadratic formula to find the values of $w$ . The quadratic formula is given by:
$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

STEP 4

Before applying the quadratic formula, we should check if the quadratic can be factored easily. If not, we will proceed with the quadratic formula. In this case, it does not seem to factor easily, so we will use the quadratic formula.

STEP 5

Identify the coefficients $a$ , $b$ , and $c$ from the standard form of the quadratic equation.
$a = 5, b = -13, c = -6$

STEP 6

Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula.
$w = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(5)(-6)}}{2(5)}$

STEP 7

Simplify the expression inside the square root (the discriminant).
$\Delta = (-13)^2 - 4(5)(-6)$
$\Delta = 169 + 120$
$\Delta = 289$

STEP 8

Now, continue simplifying the expression for $w$ .
$w = \frac{13 \pm \sqrt{289}}{10}$

STEP 9

Take the square root of the discriminant.
$\sqrt{289} = 17$

STEP 10

Substitute the value of the square root back into the expression for $w$ .
$w = \frac{13 \pm 17}{10}$

STEP 11

Solve for the two possible values of $w$ .
$w_1 = \frac{13 + 17}{10} = \frac{30}{10} = 3$
$w_2 = \frac{13 - 17}{10} = \frac{-4}{10} = -0.4$

STEP 12

The solutions to the equation $5w^2 = 13w + 6$ are $w = 3$ and $w = -0.4$ .

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