Question

which of the statements about the following quadratic equation is true?
$6x^2 - 8 = 4x^2 + 7x$
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: Rearrange to standard quadratic form

Subtract $4x^2$ and $7x$ from both sides:
$6x^2 - 4x^2 -7x -8 = 0$
$2x^2 -7x -8 = 0$

Step2: Identify coefficients

For $ax^2+bx+c=0$, $a=2$, $b=-7$, $c=-8$

Step3: Calculate discriminant

Use $\Delta = b^2 - 4ac$:
$\Delta = (-7)^2 - 4(2)(-8) = 49 + 64 = 113$

Step4: Analyze discriminant value

$\Delta = 113 > 0$, so two real roots exist.

Answer:

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