Question

which of the statements about the following quadratic equation is true? 6x² - 8 = 4x² + 7x the discriminant is greater than zero, so there are two real roots. the discriminant is greater than zero, so there are two complex roots. the discriminant is less than zero, so there are two real roots. the discriminant is less than zero, so there are two complex roots.

Explanation:

Step1: Simplify the quadratic equation

First, we need to rewrite the given quadratic equation \(6x^{2}-8 = 4x^{2}+7x\) in standard form \(ax^{2}+bx + c=0\).
Subtract \(4x^{2}\) and \(7x\) from both sides:
\(6x^{2}-4x^{2}-7x - 8=0\)
Simplify the left - hand side: \(2x^{2}-7x - 8 = 0\)
Here, \(a = 2\), \(b=-7\), \(c = - 8\).

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 \(a = 2\), \(b=-7\), \(c=-8\) into the formula:
\(D=(-7)^{2}-4\times2\times(-8)\)
First, calculate \((-7)^{2}=49\) and \(4\times2\times(-8)=-64\)
Then \(D = 49-(-64)=49 + 64=113\)

Step3: Analyze the discriminant

Since \(D = 113>0\), for a quadratic equation \(ax^{2}+bx + c = 0\), if the discriminant \(D>0\), the equation has two distinct real roots.

Answer:

The discriminant is greater than zero, so there are two real roots.