Solve the quadratic equation by factoring. (Enter your answers as a comma-separated list.)
5x2 − 23x = −12
5x^2 -23x + 12 =0
(5x - 3)(x - 4) = 0
5x -3 =0 x -4 = 0
Can you solve each of these for x?
This is not calculus. If the class is called calculus, ask for your money back.
To solve the quadratic equation by factoring, we need to rearrange the equation to have zero on one side. Let's rewrite the equation as:
5x^2 - 23x + 12 = 0
Now we need to factor the quadratic expression. We are looking for two binomials that when multiplied together, give us the quadratic expression. The factors should satisfy the following conditions:
1. The product of the first terms of the binomials should be the coefficient of x^2, which is 5x^2.
2. The product of the last terms of the binomials should be the constant term, which is 12.
3. The sum of the inner and outer terms of the binomials, when combined, should be equal to the coefficient of x, which is -23x.
To factor 5x^2 - 23x + 12, we can split the middle term (-23x) into two terms that multiply to give us the product of the leading coefficient and the constant term, which is 5 * 12 = 60, and also add up to the middle term (-23x).
Let's find two numbers that satisfy these conditions:
- The product of the numbers is 60.
- The sum of the numbers is -23.
After some trial and error, we find that -20 and -3 are the two numbers that satisfy the conditions. Let's rewrite the quadratic expression using these numbers:
5x^2 - 20x - 3x + 12 = 0
Now, we can group the terms and factor by grouping:
(5x^2 - 20x) + (-3x + 12) = 0
Factoring out the common factors from each group:
5x(x - 4) - 3(x - 4) = 0
Now, we can see that we have a common binomial factor of (x - 4):
(5x - 3)(x - 4) = 0
To find the solutions, we set each factor equal to zero and solve for x:
5x - 3 = 0 or x - 4 = 0
Solving for x in each equation:
5x = 3 or x = 4
Dividing both sides by 5:
x = 3/5 or x = 4
Therefore, the solutions to the quadratic equation 5x^2 - 23x + 12 = 0 are x = 3/5 and x = 4.