Question Solve the quadratic equation $2 d^{2} + 14 d - 16 = 0$ and select the correct solution.

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic equation in the form of $2d^2 + 14d - 16 = 0$ .
2. We are to solve for the variable $d$ .
3. The hint suggests factoring out a 2 and then dividing both sides by 2 before solving for $d$ .

STEP 2

Factor out a 2 from the quadratic equation.
$2(d^2 + 7d - 8) = 0$

STEP 3

Divide both sides of the equation by 2 to simplify the equation.
$d^2 + 7d - 8 = 0$

STEP 4

Now we need to factor the quadratic expression $d^2 + 7d - 8$ .

STEP 5

Look for two numbers that multiply to $-8$ (the constant term) and add up to $7$ (the coefficient of the middle term).

STEP 6

The numbers that satisfy these conditions are $8$ and $-1$ because $8 \times -1 = -8$ and $8 + (-1) = 7$ .

STEP 7

Rewrite the middle term using the numbers found in STEP_6.
$d^2 + 8d - d - 8 = 0$

STEP 8

Factor by grouping. Group the terms into two pairs and factor out the common factor from each pair.
$d(d + 8) - 1(d + 8) = 0$

STEP 9

Factor out the common binomial factor $(d + 8)$ .
$(d - 1)(d + 8) = 0$

STEP 10

Apply the zero-product property, which states that if a product of two factors is zero, then at least one of the factors must be zero.
$d - 1 = 0 \quad \text{or} \quad d + 8 = 0$

STEP 11

Solve each equation for $d$ .
For $d - 1 = 0$ :
$d = 1$
For $d + 8 = 0$ :
$d = -8$

STEP 12

The solutions for the variable $d$ are $d = 1$ and $d = -8$ .
Therefore, the correct answer is C) $d=1, d=-8$ .

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