5x^2+34x+24
If it factors, here is a method that will work all the time.
5x^2+34x+24
multiply the first times the last --- 5x24 = 120
now look for 2 numbers whose product is that and which add up to the middle number
120 = 30x4 and 30+4 = 34
so those numbers are 30 and 4
so rewrite the middle term as the sum of those two numbers
34x = 30x + 4x
then
5x^2+34x+24
=5x^2 + 30x + 4x + 24
=5x(x + 6) + 4(x + 6)
= (x+6)(5x+4)
x^2+34x+c
To simplify the expression 5x^2 + 34x + 24, we can first check if it can be factored. If it cannot be factored, we can use the quadratic formula to find the solutions.
1. Factoring:
To check if the expression can be factored, we need to look for two numbers that multiply to give 5 * 24 = 120 and add up to 34.
The factors of 120 are:
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Since none of these pairs add up to 34, the expression cannot be factored.
2. Quadratic Formula:
The quadratic formula can be used to find the solutions to the quadratic equation in the form ax^2 + bx + c = 0. In this case, a = 5, b = 34, and c = 24.
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values, we have:
x = (-34 ± √(34^2 - 4 * 5 * 24)) / (2 * 5)
Calculating inside the square root:
x = (-34 ± √(1156 - 480)) / 10
x = (-34 ± √676) / 10
x = (-34 ± 26) / 10
We have two solutions:
x = (-34 + 26) / 10 = -8 / 10 = -0.8
x = (-34 - 26) / 10 = -60 / 10 = -6
Therefore, the solutions to the quadratic equation 5x^2 + 34x + 24 = 0 are -0.8 and -6.