Solve Quadratic Equation d^2 = 4(3d + 7)

This problem requires solving a quadratic equation by first expanding the expression and then factoring the resulting standard form equation to find the values of d.

Answer

The problem asks to solve the quadratic equation:

Here is the step-by-step solution:

Step 1: Expand the right-hand side of the equation.

Step 2: Rearrange the equation into the standard quadratic form, .
Subtract

and

from both sides:

Step 3: Solve the quadratic equation.
We can solve this by factoring. We need two numbers that multiply to

and add up to

.
The pairs of factors of 28 are:

•1 and 28
•2 and 14
•4 and 7

Since the product is negative (

), one factor must be positive and the other negative. Since the sum is negative (

), the larger absolute value factor must be negative.
The pair

and

satisfies the conditions:

So, the quadratic equation factors as:

Step 4: Find the values of .
For the product of two factors to be zero, at least one of the factors must be zero:

Step 5: Verification (Optional).
Check

:
LHS:

RHS:

LHS = RHS. So

is a solution.

Check

:
LHS:

RHS:

LHS = RHS. So

is a solution.

The solutions are

and

The final answer is:

Solve Quadratic Equation d^2 = 4(3d + 7)

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