To solve the quadratic equation [tex]\( x^2 + 2x - 35 = 0 \)[/tex] step-by-step, we will use the quadratic formula. The general quadratic equation takes the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
In this equation:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = -35 \)[/tex]
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's break down the steps:
1. Calculate the Discriminant:
The discriminant [tex]\( \Delta \)[/tex] is calculated using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -35 \)[/tex]:
[tex]\[ \Delta = (2)^2 - 4 \cdot 1 \cdot (-35) \][/tex]
[tex]\[ \Delta = 4 + 140 \][/tex]
[tex]\[ \Delta = 144 \][/tex]
2. Calculate the Roots:
Next, use the discriminant to find the roots of the quadratic equation. The roots are given by:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\( \Delta = 144 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( a = 1 \)[/tex]:
[tex]\[ x = \frac{-2 \pm \sqrt{144}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-2 \pm 12}{2} \][/tex]
This gives us two possible solutions:
[tex]\[ x_1 = \frac{-2 + 12}{2} = \frac{10}{2} = 5 \][/tex]
[tex]\[ x_2 = \frac{-2 - 12}{2} = \frac{-14}{2} = -7 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 + 2x - 35 = 0 \)[/tex] are:
[tex]\[ x_1 = 5 \][/tex]
[tex]\[ x_2 = -7 \][/tex]
