Sure, let's solve the equation [tex]\((2x + 3)(3x - 7) = 0\)[/tex] step-by-step.
First, recognize that we have a product of two factors equal to zero. According to the zero product property, if the product of two numbers or expressions is zero, at least one of the factors must be zero.
So we set each factor equal to zero and solve for [tex]\(x\)[/tex] separately:
1. Solve the first equation: [tex]\(2x + 3 = 0\)[/tex]
[tex]\[
2x + 3 = 0
\][/tex]
- Subtract 3 from both sides:
[tex]\[
2x = -3
\][/tex]
- Divide both sides by 2:
[tex]\[
x = \frac{-3}{2}
\][/tex]
Thus, one solution is:
[tex]\[
x = -\frac{3}{2}
\][/tex]
2. Solve the second equation: [tex]\(3x - 7 = 0\)[/tex]
[tex]\[
3x - 7 = 0
\][/tex]
- Add 7 to both sides:
[tex]\[
3x = 7
\][/tex]
- Divide both sides by 3:
[tex]\[
x = \frac{7}{3}
\][/tex]
Thus, the second solution is:
[tex]\[
x = \frac{7}{3}
\][/tex]
So, the solutions to the equation [tex]\((2x + 3)(3x - 7) = 0\)[/tex] are:
[tex]\[
x = -\frac{3}{2} \quad \text{and} \quad x = \frac{7}{3}
\][/tex]
In decimal form:
[tex]\[
x = -1.5 \quad \text{and} \quad x = 2.3333333333333335
\][/tex]
