Question

which are solutions of the equation $4x^2 - 7x = 3x + 24$? check all that apply. $x = -4$ $x = -3$ $x = -\frac{3}{2}$ $x = \frac{2}{3}$ $x = 2$ $x = 4$

Explanation:

Step1: Rearrange the equation to standard quadratic form

First, we need to get all terms on one side of the equation. The given equation is \(4x^{2}-7x = 3x + 24\). Subtract \(3x\) and \(24\) from both sides to get:
\(4x^{2}-7x - 3x-24=0\)
Simplify the like - terms: \(4x^{2}-10x - 24 = 0\)
We can divide the entire equation by 2 to simplify it further: \(2x^{2}-5x - 12=0\)

Step2: Solve the quadratic equation \(2x^{2}-5x - 12 = 0\) using the quadratic formula or factoring

We will use factoring. We need to find two numbers \(a\) and \(b\) such that \(a\times b=2\times(- 12)=-24\) and \(a + b=-5\). The numbers are \(-8\) and \(3\) since \((-8)\times3=-24\) and \(-8 + 3=-5\)

Rewrite the middle term of the quadratic equation:
\(2x^{2}-8x+3x - 12 = 0\)

Factor by grouping:
\(2x(x - 4)+3(x - 4)=0\)
\((2x + 3)(x - 4)=0\)

Step3: Find the roots of the equation

Set each factor equal to zero:

• For \(2x+3 = 0\), we have \(2x=-3\), so \(x=-\frac{3}{2}\)
• For \(x - 4=0\), we have \(x = 4\)

Answer:

\(x=-\frac{3}{2}\), \(x = 4\) (i.e., the options are \(x =-\frac{3}{2}\) and \(x = 4\) or in the given option format: \(x=-\frac{3}{2}\) (the option with \(x =-\frac{3}{2}\)) and \(x = 4\) (the option with \(x = 4\)))