Question

For what real numbers x is x 2 − 10 x + 25 negative?

Answer 1

No real solutions

Explanation:

Part 1: Factoring the quadratic

The equation is in quadratic form - ax² + bx + c = 0.

Therefore, we can use a factoring technique to solve for x. I will use the quadratic formula -
`x=\frac{-b \pm \sqrt{b^(2)-4ac} }{2a}`.

`x=\frac{-(-10)\pm \sqrt{(-10)^(2)-4(1)(25)} }{2(1)}\\\\x=(10\pm√(100-4(25)) )/(2)\\\\x=(10\pm√(100-100))/(2)\\\\x=(10\pm√(0))/(2)\\\\x=(10\pm0)/(2)\\\\x=(10)/(2)\\\\\boxed{x=5}`

Part 2: Using discriminant to determine roots

Because the discriminant (square root portion of formula) was equivalent to zero, this is the only solution that proves the equation correctly. Therefore, there is no possible negative value that can be substituted for x without altering the final value that the equation is equal to.

Answer 2

no real numbers

Explanation:

x^2 − 10 x + 25

Factor

What 2 numbers multiply to 25 and add to -10

-5*-5 = 25

-5+-5 = -10

( x-5) (x-5)

This touches the graph at x =5

The parabola is positive so there are no values where the graph is negative