Question

Solve the equation for all values of x by completing the square.
5, x, squared, minus, 70, x, plus, 225, equals, 0
5x
2
−70x+225=
0

Answer 1

To solve the equation \( 5x^2 - 70x + 225 = 0 \) by completing the square, we can follow these steps:

1. **Divide the entire equation by 5** to simplify it:

\[
x^2 - 14x + 45 = 0
\]

2. **Move the constant term to the other side**:

\[
x^2 - 14x = -45
\]

3. **Complete the square on the left side**:
- Take half of the coefficient of \( x \) (which is \(-14\)), square it, and add this square to both sides. Half of \(-14\) is \(-7\), and squaring it gives \(49\).

\[
x^2 - 14x + 49 = -45 + 49
\]

This simplifies to:

\[
x^2 - 14x + 49 = 4
\]

4. **Rewrite the left side as a square**:

\[
(x - 7)^2 = 4
\]

5. **Take the square root of both sides**:

\[
x - 7 = \pm 2
\]

6. **Solve for \( x \)**:

- If \( x - 7 = 2 \):

\[
x = 9
\]

- If \( x - 7 = -2 \):

\[
x = 5
\]

Thus, the solutions to the equation are:

\[
\boxed{5 \text{ and } 9}
\]