Question

d. $2x^2 + 5x = 12$

Explanation:

Step1: Rearrange the equation

To solve the quadratic equation \(2x^{2}+5x = 12\), first, we rearrange it to the standard quadratic 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 try to factor the quadratic. 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 \(2x-3 = 0\), add 3 to both sides: \(2x=3\), then divide by 2: \(x=\frac{3}{2}\)
• For \(x + 4=0\), subtract 4 from both sides: \(x=-4\)

Answer:

The solutions are \(x=\frac{3}{2}\) and \(x = - 4\)