Question

Giselle needs help solving the following quadratic equation 5x ^ 2 + 1 = 36x - 6 Which value of x is a solution to this equation?
x = 7
x = - 7
x = - 5
x = - 1

Answer 1

To solve the quadratic equation \( 5x^2 + 1 = 36x - 6 \), we need to rearrange the equation into standard form \( ax^2 + bx + c = 0 \).

1. Move all terms to one side of the equation:

\[
5x^2 - 36x + 1 + 6 = 0
\]
\[
5x^2 - 36x + 7 = 0
\]

Now we have the quadratic equation \( 5x^2 - 36x + 7 = 0 \).

2. We can use the quadratic formula to find the solutions for \( x \):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 5 \), \( b = -36 \), and \( c = 7 \).

3. Calculate the discriminant:

\[
b^2 - 4ac = (-36)^2 - 4(5)(7) = 1296 - 140 = 1156
\]

4. Now substitute into the quadratic formula:

\[
x = \frac{-(-36) \pm \sqrt{1156}}{2(5)} = \frac{36 \pm 34}{10}
\]

5. Calculate the two potential solutions:

- For the positive case:
\[
x = \frac{36 + 34}{10} = \frac{70}{10} = 7
\]

- For the negative case:
\[
x = \frac{36 - 34}{10} = \frac{2}{10} = \frac{1}{5} \quad (\text{This value is not among the given options})
\]

So, we find that the only solution that matches one of the provided options is:

\[
\boxed{7}
\]