Question

use the discriminant to determine the number of roots of $x^2 - 3 = (3x - 2)(x + 7)$. a) one b) two c) three d) zero

Explanation:

Step1: Simplify the equation

First, expand the right - hand side of the equation \(x^{2}-3=(3x - 2)(x + 7)\).
Using the FOIL method: \((3x-2)(x + 7)=3x\times x+3x\times7-2\times x-2\times7=3x^{2}+21x-2x - 14=3x^{2}+19x-14\)
So the equation becomes \(x^{2}-3 = 3x^{2}+19x-14\)
Move all terms to one side: \(3x^{2}+19x-14-(x^{2}-3)=0\)
\(3x^{2}+19x - 14 - x^{2}+3 = 0\)
Combine like terms: \(2x^{2}+19x-11 = 0\)

Step2: Recall the discriminant formula

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a
eq0\)), the discriminant \(\Delta=b^{2}-4ac\)
In the equation \(2x^{2}+19x - 11=0\), we have \(a = 2\), \(b = 19\), \(c=-11\)

Step3: Calculate the discriminant

\(\Delta=(19)^{2}-4\times2\times(-11)\)
\(=361+88\)
\(=449\)

Step4: Determine the number of roots

Since \(\Delta = 449>0\), a quadratic equation with a positive discriminant has two distinct real roots.

Answer:

b) two