Let's solve the quadratic equation [tex]\( 2x^2 + 5x - 3 = 0 \)[/tex].
### Step-by-Step Solution
1. Identify the coefficients:
For a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
- Here, [tex]\( a = 2 \)[/tex],
- [tex]\( b = 5 \)[/tex],
- [tex]\( c = -3 \)[/tex].
2. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is:
[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]
4. Compute the square root of the discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{49} = 7
\][/tex]
5. Substitute back into the quadratic formula:
[tex]\[
x = \frac{-5 \pm 7}{2 \cdot 2}
\][/tex]
Simplify the two cases separately (using [tex]\(+\)[/tex] and [tex]\(-\)[/tex]).
Case 1 (Using +):
[tex]\[
x = \frac{-5 + 7}{4}
\][/tex]
[tex]\[
x = \frac{2}{4}
\][/tex]
[tex]\[
x = \frac{1}{2}
\][/tex]
Case 2 (Using -):
[tex]\[
x = \frac{-5 - 7}{4}
\][/tex]
[tex]\[
x = \frac{-12}{4}
\][/tex]
[tex]\[
x = -3
\][/tex]
### Final Solution
The solutions to the quadratic equation [tex]\( 2x^2 + 5x - 3 = 0 \)[/tex] are:
[tex]\[
x = -3 \quad \text{and} \quad x = \frac{1}{2}
\][/tex]
