To find the solutions to the equation [tex]\( 0 = x^2 + 5x - 6 \)[/tex] using the quadratic formula, let's walk through the given steps in detail:
### Step 1: Gather necessary information
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In the given quadratic equation [tex]\( x^2 + 5x - 6 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- [tex]\( c = -6 \)[/tex]
### Step 2: Substitute these values into the quadratic formula
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute [tex]\( a = 1 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -6 \)[/tex]:
[tex]\[ x = \frac{-5 \pm \sqrt{(5)^2 - 4(1)(-6)}}{2(1)} \][/tex]
### Step 3: Simplify the expression under the square root (the discriminant) and complete the formula
First, calculate the discriminant ([tex]\( b^2 - 4ac \)[/tex]):
[tex]\[ b^2 - 4ac = (5)^2 - 4(1)(-6) \][/tex]
[tex]\[ = 25 + 24 \][/tex]
[tex]\[ = 49 \][/tex]
Now, continue plugging the values into the quadratic formula:
[tex]\[ x = \frac{-5 \pm \sqrt{49}}{2} \][/tex]
Since the square root of 49 is 7, we get two possible solutions:
[tex]\[ x = \frac{-5 + 7}{2} \quad \text{and} \quad x = \frac{-5 - 7}{2} \][/tex]
Calculate each of these solutions:
[tex]\[ x = \frac{-5 + 7}{2} = \frac{2}{2} = 1 \][/tex]
[tex]\[ x = \frac{-5 - 7}{2} = \frac{-12}{2} = -6 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 + 5x - 6 = 0 \)[/tex] are:
[tex]\[ x = 1 \quad \text{and} \quad x = -6 \][/tex]
So, the correctly simplified and computed solutions in ascending order are:
[tex]\[ \boxed{x = -6, 1} \][/tex]
