To solve the quadratic equation [tex]\( x^2 + 3x - 28 = 0 \)[/tex], we will use the quadratic formula. The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( ax^2 + bx + c = 0 \)[/tex] is the standard form of the quadratic equation. For the given equation [tex]\( x^2 + 3x - 28 = 0 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( c = -28 \)[/tex]
Step-by-step solution:
1. Calculate the discriminant:
The discriminant [tex]\( \Delta \)[/tex] is calculated using the formula [tex]\( \Delta = b^2 - 4ac \)[/tex].
[tex]\[
\Delta = 3^2 - 4 \cdot 1 \cdot (-28)
\][/tex]
Substituting the values:
[tex]\[
\Delta = 9 + 112 = 121
\][/tex]
2. Compute the roots using the quadratic formula:
The quadratic formula provides two solutions for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Plugging the values [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( \Delta = 121 \)[/tex]:
[tex]\[
x = \frac{-3 \pm \sqrt{121}}{2 \cdot 1}
\][/tex]
3. Calculate the two solutions:
[tex]\[
x_1 = \frac{-3 + \sqrt{121}}{2} = \frac{-3 + 11}{2} = \frac{8}{2} = 4
\][/tex]
[tex]\[
x_2 = \frac{-3 - \sqrt{121}}{2} = \frac{-3 - 11}{2} = \frac{-14}{2} = -7
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 + 3x - 28 = 0 \)[/tex] are:
[tex]\[ x = 4 \][/tex]
[tex]\[ x = -7 \][/tex]
These are the two roots of the given quadratic equation.
