Question

Solve the following quadratic equation for all values of xx in simplest form.
4, left parenthesis, 10, minus, 3, x, right parenthesis, squared, minus, 4, equals, 5
4(10−3x)
2
−4=
5

Answer 1

To solve the quadratic equation \( 4(10 - 3x)^2 - 4 = 5 \), we first isolate the squared term. Here's how to do it step by step:

1. Start with the equation:
\[
4(10 - 3x)^2 - 4 = 5
\]

2. Add 4 to both sides:
\[
4(10 - 3x)^2 = 9
\]

3. Divide both sides by 4:
\[
(10 - 3x)^2 = \frac{9}{4}
\]

4. Take the square root of both sides, remembering to consider both the positive and negative roots:
\[
10 - 3x = \frac{3}{2} \quad \text{or} \quad 10 - 3x = -\frac{3}{2}
\]

5. Solve the first equation \( 10 - 3x = \frac{3}{2} \):
\[
10 - \frac{3}{2} = 3x
\]
Convert 10 to a fraction:
\[
\frac{20}{2} - \frac{3}{2} = 3x
\]
Simplifying gives:
\[
\frac{17}{2} = 3x \implies x = \frac{17}{6}
\]

6. Solve the second equation \( 10 - 3x = -\frac{3}{2} \):
\[
10 + \frac{3}{2} = 3x
\]
Again, convert 10 to a fraction:
\[
\frac{20}{2} + \frac{3}{2} = 3x
\]
Simplifying gives:
\[
\frac{23}{2} = 3x \implies x = \frac{23}{6}
\]

Thus, the solutions to the equation are:
\[
\boxed{\frac{17}{6}, \frac{23}{6}}
\]