Let’s solve the quadratic equation [tex]\(2x^2 + 7x + 3 = 0\)[/tex] step-by-step.
1. Identify the coefficients:
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a\)[/tex] is the coefficient of [tex]\(x^2\)[/tex]
- [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]
- [tex]\(c\)[/tex] is the constant term
For [tex]\(2x^2 + 7x + 3 = 0\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 7\)[/tex]
- [tex]\(c = 3\)[/tex]
2. Use the quadratic formula:
The quadratic formula to find the roots of [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Plug in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}
\][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = 7^2 - 4 \cdot 2 \cdot 3 = 49 - 24 = 25
\][/tex]
4. Compute the square root of the discriminant:
[tex]\[
\sqrt{25} = 5
\][/tex]
5. Find the two solutions:
Using the positive root:
[tex]\[
x_1 = \frac{-7 + 5}{4} = \frac{-2}{4} = -\frac{1}{2}
\][/tex]
Using the negative root:
[tex]\[
x_2 = \frac{-7 - 5}{4} = \frac{-12}{4} = -3
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(2x^2 + 7x + 3 = 0\)[/tex] are:
[tex]\[
x = -3 \quad \text{and} \quad x = -\frac{1}{2}
\][/tex]
