Question

what is the value of the discriminant of the quadratic equation (-2x^2 = -8x + 8), and what does this value mean about the number of real number solutions the equation has?

• the discriminant is equal to 0, which means the equation has no real number solutions.
• the discriminant is equal to 0, which means the equation has one real number solution.
• the discriminant is equal to 128, which means the equation has no real number solutions.
• the discriminant is equal to 128, which means the equation has two real number solutions.

Explanation:

Step1: Rewrite the equation in standard form

The standard form of a quadratic equation is \(ax^{2}+bx + c = 0\). Given the equation \(-2x^{2}=-8x + 8\), we add \(8x\) and subtract \(8\) from both sides to get: \(-2x^{2}+8x - 8=0\). We can also multiply through by \(- 1\) to make it simpler: \(2x^{2}-8x + 8 = 0\). Here, \(a = 2\), \(b=-8\), and \(c = 8\).

Step2: Calculate the discriminant

The formula for the discriminant \(D\) of a quadratic equation \(ax^{2}+bx + c = 0\) is \(D=b^{2}-4ac\). Substitute \(a = 2\), \(b=-8\), and \(c = 8\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Interpret the discriminant

For a quadratic equation \(ax^{2}+bx + c = 0\):

• If \(D>0\), the equation has two distinct real solutions.
• If \(D = 0\), the equation has one real solution (a repeated root).
• If \(D<0\), the equation has no real solutions.

Since \(D = 0\), the equation has one real number solution.

Answer:

B. The discriminant is equal to 0, which means the equation has one real number solution.