Question
x^2 -13x + 42 = 0 by grouping
Answers
GPT-4o mini
To solve the quadratic equation \( x^2 - 13x + 42 = 0 \) by grouping, we will first factor the quadratic expression.
1. First, identify two numbers that multiply to the constant term (42) and add to the coefficient of the linear term (-13).
The two numbers that satisfy both conditions are -6 and -7 since:
- \((-6) \times (-7) = 42\) (they multiply to give 42)
- \((-6) + (-7) = -13\) (they add to give -13)
2. Rewrite the quadratic expression using these two numbers:
\[
x^2 - 6x - 7x + 42 = 0
\]
3. Now group the terms:
\[
(x^2 - 6x) + (-7x + 42) = 0
\]
4. Factor each group:
\[
x(x - 6) - 7(x - 6) = 0
\]
5. Now, factor out the common factor \((x - 6)\):
\[
(x - 6)(x - 7) = 0
\]
6. Set each factor equal to zero:
\[
x - 6 = 0 \quad \text{or} \quad x - 7 = 0
\]
7. Solve for \(x\):
\[
x = 6 \quad \text{or} \quad x = 7
\]
Thus, the solutions to the equation \( x^2 - 13x + 42 = 0 \) are:
\[
x = 6 \quad \text{and} \quad x = 7
\]
1. First, identify two numbers that multiply to the constant term (42) and add to the coefficient of the linear term (-13).
The two numbers that satisfy both conditions are -6 and -7 since:
- \((-6) \times (-7) = 42\) (they multiply to give 42)
- \((-6) + (-7) = -13\) (they add to give -13)
2. Rewrite the quadratic expression using these two numbers:
\[
x^2 - 6x - 7x + 42 = 0
\]
3. Now group the terms:
\[
(x^2 - 6x) + (-7x + 42) = 0
\]
4. Factor each group:
\[
x(x - 6) - 7(x - 6) = 0
\]
5. Now, factor out the common factor \((x - 6)\):
\[
(x - 6)(x - 7) = 0
\]
6. Set each factor equal to zero:
\[
x - 6 = 0 \quad \text{or} \quad x - 7 = 0
\]
7. Solve for \(x\):
\[
x = 6 \quad \text{or} \quad x = 7
\]
Thus, the solutions to the equation \( x^2 - 13x + 42 = 0 \) are:
\[
x = 6 \quad \text{and} \quad x = 7
\]