Question

How many real roots does the quadratic function h(x) = -7x^2 + 10x - 5 have?

a. 0

b. 1

c. 2

d. 3

Answer 1

The quadratic function
`\(h(x) = -7x^2 + 10x - 5\)` has 2 real roots.

The discriminant \(Δ = -40\), being negative, indicates two complex conjugate roots. Consequently, there are no real roots for this quadratic function.

thus the correct option( c)

Step-by-step explanation:

The given quadratic function is in the form
`\(ax^2 + bx + c\)`, where
`\(a = -7\), \(b = 10\), and \(c = -5\)`. To determine the number of real roots, we can use the discriminant formula, \(Δ =
`b^2 - 4ac\)`.

For this quadratic function, the discriminant is calculated as follows:

\[Δ =
`(10)^2` - 4(-7)(-5)\]

\[Δ = 100 - 140\]

\[Δ = -40\]

Since the discriminant is negative (\(Δ < 0\)), the quadratic equation has two complex conjugate roots. The complex roots imply that there are no real roots for this quadratic function. Therefore, the correct answer is 2 (c. 2).

In a quadratic equation
`\(ax^2 + bx + c = 0\)`, the discriminant \(Δ\) determines the nature of the roots. If \(Δ > 0\), there are two distinct real roots. If \(Δ = 0\), there is one real root (a repeated root). If \(Δ < 0\), as in this case, there are two complex conjugate roots. The discriminant essentially provides insight into the positioning of the roots on the real number line.

therefore correct option is( c)

Answer 2

The correct answer is a. 0. By calculating the discriminant of the quadratic equation h(x) = -7x^2 + 10x - 5, which is negative, we can determine that the function has no real roots.

Step-by-step explanation:

The number of real roots of the quadratic function h(x) = -7x^2 + 10x - 5 can be determined by calculating the discriminant (b^2 - 4ac) from the quadratic equation ax^2 + bx + c = 0.

In this case, we have a = -7, b = 10, and c = -5. By substituting these values into the discriminant formula, we get discriminant = b^2 - 4ac = (10)^2 - 4(-7)(-5) = 100 - 140 = -40.

Since the discriminant is less than zero, it indicates that there are no real roots; the function has two complex roots.

Thus, the correct answer is a. 0.