The square of 3 less than twice a certain real number is equal to 3 less than 7 times that… — ACT

The square of 3 less than twice a certain real number is equal to 3 less than 7 times that number. If $p$ and $q$ are the two distinct real solutions to this equation with $p > q$ , what is the value of $4p - 4q$ ?

A
9
B
11
13Answer
D
14
E
1

Answer

Correctly expanding the squared expression leads to the quadratic equation

4x^2 - 19x + 12 = 0

. Factoring this equation yields the solutions

4

and

3/4

. Since the problem specifies

p > q

, we have

p = 4

and

q = 3/4

. Substituting these values into the expression

4p - 4q

gives

4(4) - 4(3/4) = 16 - 3 = 13

Step-by-Step Solution

Translate the verbal description into an algebraic equation.

(2x - 3)^2 = 7x - 3

'Twice a number' is represented as

2x

, '3 less than twice the number' is

2x - 3

, and 'the square of' that is

(2x - 3)^2

. This is set equal to '3 less than 7 times that number', which is

7x - 3

Expand the binomial and write the equation in standard quadratic form

ax^2 + bx + c = 0

4x^2 - 19x + 12 = 0

Expanding the left side yields

4x^2 - 12x + 9 = 7x - 3

. Subtracting

7x

and adding

3

to both sides results in the standard form.

Factor the quadratic equation by grouping.

(4x - 3)(x - 4) = 0

We need two numbers that multiply to

4 \times 12 = 48

and add to

-19

. These numbers are

-16

and

-3

. Rewriting the equation as

4x^2 - 16x - 3x + 12 = 0

allows us to factor out

4x(x - 4) - 3(x - 4) = 0

, which simplifies to

(4x - 3)(x - 4) = 0

Find the roots of the equation and evaluate the target expression

4p - 4q

p = 4

q = 3/4

, and the final value is

13

Setting each factor to zero gives

x = 3/4

and

x = 4

. Since

p > q

, we assign

p = 4

and

q = 3/4

. Evaluating the expression yields

4(4) - 4(3/4) = 16 - 3 = 13

Key Concept

Solving Quadratic Equations by Factoring

Estimated Time:2m 30s

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