Question

question 13
what are the solutions to this equation?
$2x^2 - 5x = 12$
answer
solution #1: $x = $
solution #2: $x = $
submit answer

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the quadratic equation \(2x^{2}-5x = 12\) in standard form \(ax^{2}+bx + c=0\). Subtract 12 from both sides:
\(2x^{2}-5x - 12=0\)

Step2: Factor the quadratic (or use quadratic formula)

We can factor the quadratic or use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). For \(2x^{2}-5x - 12 = 0\), \(a = 2\), \(b=- 5\), \(c=-12\).
First, let's try factoring. We need two numbers that multiply to \(2\times(-12)=-24\) and add up to \(-5\). The numbers are \(-8\) and \(3\).
Rewrite the middle term:
\(2x^{2}-8x + 3x-12 = 0\)
Group the terms:
\((2x^{2}-8x)+(3x - 12)=0\)
Factor out the common factors from each group:
\(2x(x - 4)+3(x - 4)=0\)
Now, factor out \((x - 4)\):
\((2x + 3)(x - 4)=0\)

Step3: Solve for x

Set each factor equal to zero:

• For \(2x+3 = 0\):

Subtract 3 from both sides: \(2x=-3\)
Divide by 2: \(x =-\frac{3}{2}\)

• For \(x - 4=0\):

Add 4 to both sides: \(x = 4\)

Answer:

Solution #1: \(x = 4\)
Solution #2: \(x=-\frac{3}{2}\)