Question
METHOD I: FACTORING 1. \( x^{2}-6 x-40=0 \) \[ 2\left(x^{2}-3 x-20\right) \]
Ask by Turner Matthews. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \( x = -4 \) or \( x = 10 \).
Solution
**Step 1. Write Down the Equation**
We start with the quadratic equation:
\[
x^2-6x-40=0
\]
**Step 2. Find Two Numbers**
We need to find two numbers that multiply to \(-40\) (the constant term) and add to \(-6\) (the coefficient of \(x\)).
After testing a few pairs, we find that the numbers \(4\) and \(-10\) work because:
\[
4 \times (-10) = -40 \quad \text{and} \quad 4 + (-10) = -6
\]
**Step 3. Factor the Quadratic**
Using these numbers, we can factor the quadratic as follows:
\[
x^2-6x-40=(x+4)(x-10)
\]
**Step 4. Set Each Factor Equal to Zero**
For the product to be zero, at least one of the factors must equal zero. Thus, we set:
\[
x+4=0 \quad \text{or} \quad x-10=0
\]
**Step 5. Solve for \(x\)**
Solving each equation gives:
For \(x+4=0\):
\[
x = -4
\]
For \(x-10=0\):
\[
x = 10
\]
**Conclusion**
The solutions to the equation are:
\[
x=-4 \quad \text{or} \quad x=10
\]
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Simplify this solution The Deep Dive
Fun fact: This quadratic equation can be solved using factoring because it can be expressed as the product of two binomials. The numbers you are looking for are those that multiply to -40 (the constant term) and add to -6 (the coefficient of x). In this case, \( -10 \) and \( 4 \) fit the bill since \( -10 \cdot 4 = -40 \) and \( -10 + 4 = -6 \). So, we can express \( x^2 - 6x - 40 \) as \( (x - 10)(x + 4) = 0 \). To bring the equation to life, set each factor equal to zero! This results in the solutions \( x = 10 \) and \( x = -4 \).
