I need help to solve this problem by factoring: 2x^2-x-3=0
(2x-3)(x+1)
2x^2-x-3=0
first step: 2*2x^-x-3=0
4x^-x-3=0
(4x^-4x)+(3x-3)=0
4x(x-1)+3(x-1)
(4x+3)(x-1)=0
(4x+3)=0 (x-1)=0
4x+3=0 X-1=0
-3 -3 -1 -1
4x=-3 x=-1
x=-3/4
Therefore, X=(-3/4, -1)
To factor a quadratic equation, you need to find two binomials that, when multiplied, equal the quadratic equation. Let's solve the problem step by step:
1. Write down the equation: 2x^2 - x - 3 = 0.
2. Multiply the coefficient of the x^2 term by the constant term. In this case, it would be 2 * -3 = -6.
3. Find two numbers whose product is equal to -6 and whose sum is equal to the coefficient of the x term (-1). In this case, the numbers are -3 and 2, since -3 * 2 = -6 and -3 + 2 = -1.
4. Rewrite the middle term (-x) as the sum of these two numbers. Replace -x with -3x + 2x. The equation becomes:
2x^2 - 3x + 2x - 3 = 0
5. Group the terms together and factor by grouping. From the previous step, you have:
(x^2 - 3x) + (2x - 3) = 0
6. Factor out the greatest common factor from each group. In this case, x is the greatest common factor, so you can factor it out, resulting in:
x(x - 3) + 1(2x - 3) = 0
7. Simplify the expression further:
x(x - 3) + 1(2x - 3) = 0
x(x - 3) + 2x - 3 = 0
8. Now, you have two binomials: (x - 3) and (2x - 3).
Therefore, factoring the quadratic equation 2x^2 - x - 3 = 0 yields:
(x - 3)(2x - 1) = 0
To find the solutions, set each binomial equal to zero and solve for x:
x - 3 = 0 => x = 3
2x - 1 = 0 => 2x = 1 => x = 1/2
So, the solutions to the quadratic equation are x = 3 and x = 1/2.