Fully factorize x² - 14x - 32
Understand the Problem
The question is asking to fully factor the quadratic expression x² - 14x - 32. This involves finding two binomials that, when multiplied, give back the original expression.
Answer
The fully factored form is \( (x - 16)(x + 2) \).
Answer for screen readers
The fully factored form of ( x^2 - 14x - 32 ) is ( (x - 16)(x + 2) ).
Steps to Solve
- Identify the quadratic expression
We start with the quadratic expression ( x^2 - 14x - 32 ).
- Find factors of the constant term
We need to find two numbers that multiply to the constant term (-32) and add up to the coefficient of the linear term (-14).
The target numbers are factors of -32. They need to satisfy:
- ( a \cdot b = -32 )
- ( a + b = -14 )
After testing pairs of factors, we find that:
- ( -16 ) and ( 2 ) work, since: $$ -16 \cdot 2 = -32 $$ and $$ -16 + 2 = -14 $$
- Write the factors as binomials
Using the numbers found, we can rewrite the quadratic expression in factored form: $$ x^2 - 14x - 32 = (x - 16)(x + 2) $$
- Verify the factorization
We can check if our factorization is correct by expanding the binomials back: $$ (x - 16)(x + 2) = x^2 + 2x - 16x - 32 = x^2 - 14x - 32 $$
This confirms our factorization is accurate.
The fully factored form of ( x^2 - 14x - 32 ) is ( (x - 16)(x + 2) ).
More Information
This factorization shows that the roots of the equation ( x^2 - 14x - 32 = 0 ) are ( x = 16 ) and ( x = -2 ). Factoring quadratics is a useful method in algebra for solving equations efficiently.
Tips
- Forgetting to check the factor pairs properly; make sure both conditions (product and sum) are satisfied.
- Miscalculating the addition or multiplication of the factors.
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