Solve the following equation by factoring.
Rewrite the equation in factored form.
(Type your answer in factored form.)
The Correct Answer and Explanation is:
Answer:
(x+5)(x-8)
Explanation:
To solve the quadratic equation x² = 3x + 40 by factoring, the first step is to rearrange the equation into the standard quadratic form, which is ax² + bx + c = 0. This is accomplished by moving all terms to one side of the equation, setting the expression equal to zero.
Starting with the given equation:
x² = 3x + 40
Subtract 3x and 40 from both sides to bring all terms to the left side:
x² – 3x – 40 = 0
Now that the equation is in standard form, the next step is to factor the trinomial x² – 3x – 40. The goal is to find two numbers that simultaneously satisfy two conditions:
- They must multiply to the constant term, which is -40.
- They must add up to the coefficient of the middle term (the x-term), which is -3.
Consider the pairs of integers that multiply to -40 and check their sum:
- 1 and -40 (Sum = -39)
- 2 and -20 (Sum = -18)
- 4 and -10 (Sum = -6)
- 5 and -8 (Sum = -3)
The pair 5 and -8 meets both requirements, as their product is -40 and their sum is -3. These two numbers are used to write the trinomial as a product of two binomials.
The factored form of the expression is (x + 5)(x – 8).
Therefore, the equation rewritten in factored form is:
(x + 5)(x – 8) = 0
The expression (x + 5)(x – 8) is the correct answer for the box. The order of the factors can be reversed to (x – 8)(x + 5) and still be correct.