Question

question 15 (mandatory) (1 point)
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: Expand the right - hand side

First, we expand \((3x - 2)(x + 7)\) using the distributive property (FOIL method).
\((3x-2)(x + 7)=3x\times x+3x\times7-2\times x - 2\times7=3x^{2}+21x-2x - 14=3x^{2}+19x - 14\)

Step2: Rewrite the equation in standard form

The original equation is \(x^{2}-3=(3x - 2)(x + 7)\). Substitute the expanded form of the right - hand side into the equation:
\(x^{2}-3 = 3x^{2}+19x-14\)
Move all terms to one side to get the standard quadratic form \(ax^{2}+bx + c = 0\).
\(3x^{2}+19x-14-(x^{2}-3)=0\)
\(3x^{2}+19x-14 - x^{2}+3 = 0\)
Combine like terms: \((3x^{2}-x^{2})+19x+(-14 + 3)=0\)
\(2x^{2}+19x-11 = 0\)

Step3: Identify \(a\), \(b\), and \(c\)

For a quadratic equation \(ax^{2}+bx + c = 0\), in the equation \(2x^{2}+19x-11 = 0\), we have \(a = 2\), \(b = 19\), and \(c=-11\).

Step4: Calculate the discriminant

The discriminant of a quadratic equation \(ax^{2}+bx + c = 0\) is given by the formula \(\Delta=b^{2}-4ac\).
Substitute \(a = 2\), \(b = 19\), and \(c=-11\) into the formula:
\(\Delta=(19)^{2}-4\times2\times(-11)\)
\(\Delta = 361+88\)
\(\Delta=449\)

Step5: Determine the number of roots based on the discriminant

Since the discriminant \(\Delta=b^{2}-4ac>0\) (because \(449>0\)), a quadratic equation \(ax^{2}+bx + c = 0\) has two distinct real roots when \(\Delta>0\).

Answer:

b) two