Math

Question Solve for qq where the product of two linear expressions equals 0. Write the solutions as integers or simplified fractions.

Studdy Solution

STEP 1

Assumptions
1. We have a quadratic equation in the form of ((aq+b)(cq+d))=0((a q + b)(c q + d)) = 0.
2. The values of a,b,c,a, b, c, and dd are given as a=8,b=7,c=7,a = -8, b = 7, c = 7, and d=3d = 3.
3. We need to solve for qq.
4. The Zero Product Property states that if a product of two factors is zero, then at least one of the factors must be zero.

STEP 2

Write down the quadratic equation using the given values.
((8q+7)(7q+3))=0((-8 q + 7)(7 q + 3)) = 0

STEP 3

Apply the Zero Product Property, which states that if the product of two expressions is zero, then at least one of the expressions must be zero.
8q+7=0or7q+3=0-8 q + 7 = 0 \quad \text{or} \quad 7 q + 3 = 0

STEP 4

Solve the first equation for qq.
8q+7=0-8 q + 7 = 0

STEP 5

Subtract 7 from both sides of the equation.
8q=7-8 q = -7

STEP 6

Divide both sides of the equation by 8-8 to isolate qq.
q=78q = \frac{-7}{-8}

STEP 7

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1 in this case.
q=78q = \frac{7}{8}

STEP 8

Now, solve the second equation for qq.
7q+3=07 q + 3 = 0

STEP 9

Subtract 3 from both sides of the equation.
7q=37 q = -3

STEP 10

Divide both sides of the equation by 77 to isolate qq.
q=37q = \frac{-3}{7}

STEP 11

The fraction 37\frac{-3}{7} is already in its simplest form, as 3 and 7 are relatively prime.

STEP 12

We have found the two solutions for qq.
q=78orq=37q = \frac{7}{8} \quad \text{or} \quad q = \frac{-3}{7}
These are the solutions in the form of proper fractions in simplest form.

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