Sure, let's solve the quadratic equation [tex]\(x^2 + 4x - 10 = 0\)[/tex] using the quadratic formula step-by-step. The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here are the coefficients for our quadratic equation [tex]\(x^2 + 4x - 10 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 4\)[/tex]
- [tex]\(c = -10\)[/tex]
### Step 1: Calculate the Discriminant
The discriminant [tex]\(\Delta\)[/tex] of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 4^2 - 4 \cdot 1 \cdot (-10) \][/tex]
[tex]\[ \Delta = 16 + 40 \][/tex]
[tex]\[ \Delta = 56 \][/tex]
The discriminant is [tex]\(56\)[/tex].
### Step 2: Determine the Nature of the Roots
Since the discriminant [tex]\(\Delta = 56\)[/tex] is greater than zero, this means that the equation has two distinct real roots.
### Step 3: Apply the Quadratic Formula
We will use the quadratic formula to find the two roots.
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(b\)[/tex], [tex]\(\Delta\)[/tex], and [tex]\(a\)[/tex]:
[tex]\[ x = \frac{-4 \pm \sqrt{56}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{56}}{2} \][/tex]
### Step 4: Simplify the Roots
Simplify the expression under the square root and then further simplify the entire expression:
[tex]\[ \sqrt{56} = \sqrt{4 \cdot 14} = 2\sqrt{14} \][/tex]
Therefore:
[tex]\[ x = \frac{-4 \pm 2\sqrt{14}}{2} \][/tex]
[tex]\[ x = \frac{-4}{2} \pm \frac{2\sqrt{14}}{2} \][/tex]
[tex]\[ x = -2 \pm \sqrt{14} \][/tex]
Thus, the solutions are:
[tex]\[ x_1 = -2 + \sqrt{14} \][/tex]
[tex]\[ x_2 = -2 - \sqrt{14} \][/tex]
So, the correct solution to the quadratic equation [tex]\(x^2 + 4x - 10 = 0\)[/tex] is:
[tex]\[ x = -2 + \sqrt{14}, -2 - \sqrt{14} \][/tex]
Therefore, the correct choice is:
A. [tex]\[x = -2 + \sqrt{14}, -2 - \sqrt{14}\][/tex]
