Q. Which is a solution to \( (x – 2)(x + 10) = 13 \)?
Answer
Expand and solve the equation:
\[
(x-2)(x+10)=x^2+8x-20
\]
Set this equal to 13:
\[
x^2+8x-20=13
\]
Bring all terms to one side:
\[
x^2+8x-33=0
\]
Compute the discriminant:
\[
D=b^2-4ac=8^2-4\cdot1\cdot(-33)=64+132=196=14^2
\]
Apply the quadratic formula:
\[
x=\dfrac{-b\pm\sqrt{D}}{2a}=\dfrac{-8\pm14}{2}
\]
Thus
\[
x=\dfrac{-8+14}{2}=3\quad\text{or}\quad x=\dfrac{-8-14}{2}=-11
\]
Final answer: \(\boxed{3,\;-11}\)
Detailed Explanation
Problem: Solve: \( (x-2)(x+10)=13 \)
Step 1 – Expand the left side
Multiply the binomials:
\[
(x-2)(x+10)=x\cdot x + x\cdot 10 + (-2)\cdot x + (-2)\cdot 10
= x^{2}+10x-2x-20
\]
Step 2 – Combine like terms
\[
x^{2}+10x-2x-20 = x^{2}+8x-20
\]
Step 3 – Move all terms to one side
\[
x^{2}+8x-20 = 13
\]
\[
x^{2}+8x-20-13 = 0 \quad\text{so}\quad x^{2}+8x-33 = 0
\]
Step 4 – Factor the quadratic
Find two numbers whose product is \(-33\) and whose sum is \(8\): these are \(11\) and \(-3\). Factor:
\[
x^{2}+8x-33 = (x+11)(x-3)
\]
Step 5 – Solve using the Zero Product Property
\[
(x+11)(x-3)=0
\]
Set each factor equal to zero:
\[
x+11=0 \quad\text{so}\quad x=-11
\]
\[
x-3=0 \quad\text{so}\quad x=3
\]
Answer
\[
x=-11 \quad\text{or}\quad x=3
\]
FAQs
Which values of x satisfy ( (x-2)(x+10)=13 )?
How do I rearrange the equation to standard quadratic form?
Can (x^2+8x-33) be factored?
How would I use the quadratic formula here?
What does the discriminant tell me about the roots?
How can I check if a proposed number is a solution?
How does the graph help find solutions?
Are the solutions integers, rational, or real?
Choose one you love!
Math, Calculus, Geometry, etc.
Unlimited AI homework help 
Diagram Generator, Flashcard Maker, Notes Generator, Research Assistant, Answer Generator, AI Homework Helper & AI Detector