Solve the equation by factoring.
5x^2+9x-72=0
To solve the equation by factoring, we want to find two numbers that multiply to give the constant term (in this case -72) and add up to give the coefficient of the middle term (in this case 9).
The two numbers that satisfy these conditions are 12 and -6.
Therefore, we rewrite the equation as:
5x^2 + 12x - 6x - 72 = 0.
Now we factor by grouping:
(5x^2 + 12x) - (6x + 72) = 0
x(5x + 12) - 6(5x + 12) = 0
(x - 6)(5x + 12) = 0
Now we can set each factor equal to zero and solve for x:
x - 6 = 0 or 5x + 12 = 0
x = 6 or 5x = -12
x = -12/5
So the solutions to the equation are x = 6 and x = -12/5.
To solve the equation 5x^2 + 9x - 72 = 0 by factoring, we need to find two numbers whose product is -360 (the product of the coefficient of x^2 and the constant term -72) and whose sum is the coefficient of x, which is 9.
Let's break down -360 into its factors:
-360 = -1 × 2 × 2 × 2 × 3 × 3 × 5
Now, we need to find two numbers that add up to 9. Looking at the factors, we can see that the pair of numbers that met this condition are 15 and -24.
So, we can rewrite the equation as follows:
5x^2 + 15x - 24x - 72 = 0
Now, factor by grouping:
(5x^2 + 15x) - (24x + 72) = 0
5x(x + 3) - 24(x + 3) = 0
Notice that both terms have a common factor of (x + 3):
(x + 3)(5x - 24) = 0
Set each factor equal to zero and solve for x:
x + 3 = 0 or 5x - 24 = 0
For x + 3 = 0:
x = -3
For 5x - 24 = 0:
5x = 24
x = 24/5
Therefore, the solutions to the equation 5x^2 + 9x - 72 = 0 are x = -3 and x = 24/5.