how many real number solutions are ther to the equation 10x^2+31x=-24
To determine the number of real number solutions to the equation 10x^2 + 31x = -24, we need to solve the equation and observe the roots.
Step 1: Rewrite the equation to bring all terms to one side:
10x^2 + 31x + 24 = 0
Step 2: Attempt to factor the quadratic equation. In this case, it is not possible to factorize the equation, so we move on to the next step.
Step 3: Use the quadratic formula to find the solutions. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 10, b = 31, and c = 24. Inserting these values into the formula, we get:
x = (-31 ± √(31^2 - 4 * 10 * 24)) / (2 * 10)
Step 4: Simplify the equation:
x = (-31 ± √(961 - 960)) / 20
x = (-31 ± √1) / 20
x = (-31 ± 1) / 20
So, the two solutions are x = (-31 + 1) / 20 and x = (-31 - 1) / 20.
Simplifying further, we have x = -30/20 and x = -32/20, which can be simplified as x = -3/2 and x = -8/5.
Thus, the equation 10x^2 + 31x = -24 has two real number solutions: x = -3/2 and x = -8/5.