I need to know how to solve quadratic equations by factoring. Can you so a step by step explanation to this sample problem?
2
x -12x-28
x^2-12x-28=0
(x-14)(x+2)=0
x-14=0 or x+2=0
x=14 or x=-2
Sure! To solve quadratic equations by factoring, you need to find two numbers that multiply to give you the constant term (in this case, -28), and add or subtract to give you the coefficient of the x term (in this case, -12).
Let's break it down step by step:
1. Write out the quadratic equation: 2x^2 - 12x - 28 = 0.
2. Look at the coefficient of the x^2 term, which is 2. If it's not 1, you'll want to factor it out. However, in this case, 2 is already factored.
3. Examine the constant term, which is -28. We need to find two numbers that multiply to -28. In this case, -4 and 7 would work because (-4) * 7 = -28.
4. Now, we need to find two numbers that add or subtract to give us the coefficient of the x term, which is -12. In this case, -4 and 7 would work because (-4) + 7 = -12.
5. Rewrite the middle term, -12x, as the sum of -4x and -8x. Our quadratic equation now looks like this: 2x^2 - 4x - 8x - 28 = 0.
6. Group the terms: (2x^2 - 4x) + (-8x - 28) = 0.
7. Factor out the common terms from each group: 2x(x - 2) - 4(x - 2) = 0.
8. We now have a common factor of (x - 2) in both terms: (x - 2)(2x - 4) = 0.
9. To solve for x, we set each factor equal to zero and solve for x separately:
- (x - 2) = 0
- (2x - 4) = 0
10. Solving each equation:
- x - 2 = 0
Adding 2 to both sides: x = 2
- 2x - 4 = 0
Adding 4 to both sides and dividing by 2: x = 2
11. So, x = 2 is the solution to the quadratic equation 2x^2 - 12x - 28 = 0.
That's it! You have successfully solved the quadratic equation using factoring.