Question

use the discriminant to answer the questions.
32x - 4 = 4x² + 60
for the equation shown, choose the description of
the solutions.
one real, rational
two real, rational
two real, irrational
two complex

Explanation:

Step 1: Rewrite the equation in standard form

First, we rewrite the given equation \(32x - 4 = 4x^2 + 60\) into the standard quadratic form \(ax^2 + bx + c = 0\).

Subtract \(32x\) and add \(4\) to both sides:
\(4x^2 - 32x + 64 = 0\)

We can simplify this equation by dividing all terms by \(4\):
\(x^2 - 8x + 16 = 0\)

So, \(a = 1\), \(b = -8\), and \(c = 16\).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(\Delta = b^2 - 4ac\).

Substitute \(a = 1\), \(b = -8\), and \(c = 16\) into the formula:
\(\Delta = (-8)^2 - 4(1)(16)\)
\(\Delta = 64 - 64\)
\(\Delta = 0\)

Step 3: Analyze the discriminant

• If \(\Delta > 0\), the equation has two distinct real roots. If \(\Delta\) is a perfect square, the roots are rational; otherwise, they are irrational.
• If \(\Delta = 0\), the equation has one real root (a repeated root, which is rational).
• If \(\Delta < 0\), the equation has two complex conjugate roots.

Since \(\Delta = 0\), the equation has one real, rational solution (because the root will be \(\frac{-b}{2a}=\frac{8}{2}=4\), which is rational).

Answer:

One real, rational