Without​ graphing, determine how many​ x-intercepts the function has.

Question

y=-x^2+3x+10

Answer 1

To find the x-intercepts of a quadratic function, we need to solve the equation y = -x^2 + 3x + 10 for when y = 0.

Set y equal to zero:
0 = -x^2 + 3x + 10

Now we have a quadratic equation. To solve it, we can either use factoring, completing the square, or the quadratic formula.

To determine the number of x-intercepts, we can examine the discriminant of the quadratic equation, which is the value under the square root in the quadratic formula.

The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.

In our equation -x^2 + 3x + 10, a = -1, b = 3, and c = 10, so the discriminant is:
3^2 - 4(-1)(10) = 9 + 40 = 49

Since the discriminant is positive (49 > 0), this means the quadratic equation has two distinct solutions, and therefore, the function has 2 x-intercepts.