Connect with knowledgeable individuals and get your questions answered on IDNLearn.com. Ask your questions and receive prompt, detailed answers from our experienced and knowledgeable community members.

What are the two solutions?

[tex]\[
\begin{array}{l}
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)} \\
x = \frac{2 \pm 6}{2}
\end{array}
\][/tex]

A. [tex]\(x = 4, x = -4\)[/tex]
B. [tex]\(x = 3, x = 4\)[/tex]
C. [tex]\(x = 6, x = -6\)[/tex]
D. [tex]\(x = 4, x = -2\)[/tex]


Sagot :

To find the solutions for the quadratic equation given by

[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)} \][/tex]

we go through the following steps:

1. Identify the coefficients: For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we have:
[tex]\[ a = 1, \quad b = -2, \quad c = -8 \][/tex]

2. Calculate the discriminant: The discriminant [tex]\(\Delta\)[/tex] of the quadratic equation is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-2)^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36 \][/tex]

3. Find the square root of the discriminant:
[tex]\[ \sqrt{\Delta} = \sqrt{36} = 6 \][/tex]

4. Apply the quadratic formula: The roots [tex]\(x\)[/tex] are given by:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
With [tex]\(b = -2\)[/tex], [tex]\(a = 1\)[/tex], and [tex]\(\sqrt{\Delta} = 6\)[/tex]:
[tex]\[ x = \frac{-(-2) \pm 6}{2 \cdot 1} = \frac{2 \pm 6}{2} \][/tex]

This gives us two possible solutions:
[tex]\[ x_1 = \frac{2 + 6}{2} = \frac{8}{2} = 4 \][/tex]
[tex]\[ x_2 = \frac{2 - 6}{2} = \frac{-4}{2} = -2 \][/tex]

Therefore, the two solutions for the quadratic equation are:
[tex]\[ x = 4 \quad \text{and} \quad x = -2 \][/tex]

So, the correct answer is:
[tex]\[ x = 4 ; x = -2 \][/tex]