Let's solve the quadratic equation [tex]\(2x^2 - 5x - 3 = 0\)[/tex] step-by-step.
1. Identify the coefficients:
For the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[a = 2\][/tex]
[tex]\[b = -5\][/tex]
[tex]\[c = -3\][/tex]
2. Calculate the discriminant:
The discriminant (Δ) is given by the formula:
[tex]\[\Delta = b^2 - 4ac\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[\Delta = (-5)^2 - 4 \cdot 2 \cdot (-3)\][/tex]
[tex]\[\Delta = 25 + 24\][/tex]
[tex]\[\Delta = 49\][/tex]
3. Determine the nature of the roots:
Since the discriminant is positive (Δ > 0), there are two distinct real roots.
4. Find the roots using the quadratic formula:
The quadratic formula is given by:
[tex]\[x = \frac{-b \pm \sqrt{\Delta}}{2a}\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[x = \frac{-(-5) \pm \sqrt{49}}{2 \cdot 2}\][/tex]
[tex]\[x = \frac{5 \pm 7}{4}\][/tex]
Calculate each root separately:
[tex]\[
x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3
\][/tex]
[tex]\[
x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}
\][/tex]
Therefore, the roots of the equation are:
[tex]\[x_1 = 3\][/tex]
[tex]\[x_2 = -\frac{1}{2}\][/tex]
5. Check the given options:
- A. [tex]\(x = -\frac{1}{2}\)[/tex]: This is one of the solutions.
- B. [tex]\(x = -3\)[/tex]: This is not a solution.
- C. [tex]\(x = 3\)[/tex]: This is one of the solutions.
- D. [tex]\(x = \frac{1}{2}\)[/tex]: This is not a solution.
- E. [tex]\(x = 2\)[/tex]: This is not a solution.
6. Correct options:
The correct answers are A ([tex]\(x = -\frac{1}{2}\)[/tex]) and C ([tex]\(x = 3\)[/tex]).
