Solve the equation *x*2 − 18*x* = 40.

A.

*x* = −15 and *x* = −3

B.

*x* = 20 and *x* = −2

C.

*x* = 10 and *x* = 4

D.

*x* = −20 and *x* = −2

To solve the quadratic equation \(x^2 - 18x = 40\), we first need to rewrite it in standard form:

\[
x^2 - 18x - 40 = 0
\]

Now, we can solve this equation using the quadratic formula, which is:

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

For the equation \(x^2 - 18x - 40 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -18\)
- \(c = -40\)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(1)(-40)}}{2(1)}
\]

Simplify the expression:

\[
x = \frac{18 \pm \sqrt{324 + 160}}{2}
\]

\[
x = \frac{18 \pm \sqrt{484}}{2}
\]

\[
x = \frac{18 \pm 22}{2}
\]

Now, solve for the two possible values of \(x\):

1. \(x = \frac{18 + 22}{2} = \frac{40}{2} = 20\)
2. \(x = \frac{18 - 22}{2} = \frac{-4}{2} = -2\)

So the solutions are \(x = 20\) and \(x = -2\).

**Correct choice:**
- B. \( x = 20 \) and \( x = -2 \)

Would you like further explanation or have any questions? Here are some related problems to try:

1. How do you factor a quadratic equation when the quadratic formula is not required?
2. Can you derive the quadratic formula starting from the general form \(ax^2 + bx + c = 0\)?
3. What is the discriminant of the quadratic equation \(x^2 - 18x - 40 = 0\) and what does it tell you?
4. How can completing the square be used to solve the quadratic equation \(x^2 - 18x = 40\)?
5. How would the solution change if the equation were \(x^2 - 18x + 40 = 0\)?

**Tip:** When solving quadratic equations, double-check your discriminant calculation to ensure it determines the correct nature of the roots!

Learn how to solve the quadratic equation x^2 - 18x = 40 with a detailed step-by-step solution using the quadratic formula. This problem covers key algebraic concepts such as quadratic equations and their solutions. Suitable for students in Grades 9-12.

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