Solve the following quadratic equation by factoring. 1. 2x^2 – 5x – 18 = 0 2. x^2 + 6x = 16 Please help!
The Correct Answer and Explanation is:
1. Solve:
2x² – 5x – 18 = 0
Step 1: Multiply the coefficient of x² (which is 2) by the constant term (which is -18)
2 × (-18) = -36
Step 2: Find two numbers that multiply to -36 and add to -5
The numbers are -9 and 4 because:
-9 × 4 = -36
-9 + 4 = -5
Step 3: Rewrite the middle term using these numbers
2x² – 9x + 4x – 18 = 0
Step 4: Group terms and factor
(2x² – 9x) + (4x – 18) = 0
x(2x – 9) + 2(2x – 9) = 0
Step 5: Factor out the common binomial
(2x – 9)(x + 2) = 0
Step 6: Solve for x
2x – 9 = 0 or x + 2 = 0
First solution:
2x = 9
x = 9/2
Second solution:
x = -2
Final Answer:
x = 9/2 or x = -2
2. Solve:
x² + 6x = 16
Step 1: Bring all terms to one side
x² + 6x – 16 = 0
Step 2: Find two numbers that multiply to -16 and add to 6
The numbers are 8 and -2 because:
8 × (-2) = -16
8 + (-2) = 6
Step 3: Rewrite the equation
x² + 8x – 2x – 16 = 0
Step 4: Group terms and factor
(x² + 8x) – (2x + 16) = 0
x(x + 8) – 2(x + 8) = 0
Step 5: Factor out the common binomial
(x + 8)(x – 2) = 0
Step 6: Solve for x
x + 8 = 0 or x – 2 = 0
First solution:
x = -8
Second solution:
x = 2
Final Answer:
x = -8 or x = 2
Explanation Summary:
Factoring is a method for solving quadratic equations by expressing them as a product of two binomials set equal to zero. Once factored, the Zero Product Property tells us that if the product of two expressions is zero, then at least one of them must be zero. This leads to simple linear equations that are easy to solve for x. The key step is to find two numbers that multiply to the product of the leading coefficient and the constant term while also adding to the middle coefficient. Rearranging and grouping terms allows for factoring by grouping, revealing the solutions. This method is reliable for many quadratic equations when factoring is possible.
