Question

question 10 of 10
find the solutions to the equation below.
check all that apply.
4x² - 4x - 3 = 0
a. x = 3/2
b. x = 1/2
c. x = 2
d. x = 3
e. x = -1/2
f. x = -2

Explanation:

Step1: Factor the quadratic equation

We use the AC - method. For the quadratic equation \(ax^{2}+bx + c=0\) (here \(a = 4\), \(b=-4\), \(c = - 3\)), we need to find two numbers that multiply to \(ac=4\times(-3)=-12\) and add up to \(b=-4\). The numbers are \(-6\) and \(2\).
\[4x^{2}-4x - 3=4x^{2}-6x+2x - 3=2x(2x - 3)+1(2x - 3)=(2x - 3)(2x+1)=0\]

Step2: Solve for \(x\) using the zero - product property

If \((2x - 3)(2x + 1)=0\), then either \(2x-3 = 0\) or \(2x + 1=0\).
For \(2x-3=0\), we add \(3\) to both sides: \(2x=3\), then \(x=\frac{3}{2}\).
For \(2x + 1=0\), we subtract \(1\) from both sides: \(2x=-1\), then \(x=-\frac{1}{2}\).

Answer:

A. \(x=\frac{3}{2}\)
E. \(x=-\frac{1}{2}\)