Which statement is true for the equation $6 x^2-8=4 x^2+7 x$?

AThe discriminant is less than zero, so there are two complex roots.
BThe discriminant is greater than zero, so there are two real roots.
CThe discriminant is less than zero, so there are two real roots.
DThe discriminant is greater than zero, so there are two complex roots.

Question

02-11-2026
Mathematics

Answered

Which statement is true for the equation $6 x^2-8=4 x^2+7 x$?

AThe discriminant is less than zero, so there are two complex roots.
BThe discriminant is greater than zero, so there are two real roots.
CThe discriminant is less than zero, so there are two real roots.
DThe discriminant is greater than zero, so there are two complex roots.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T14:56:43-05:00

Rearrange the equation to the standard form $2x^2 - 7x - 8 = 0$.
Identify coefficients $A=2$, $B=-7$, and $C=-8$.
Calculate the discriminant using $D = B^2 - 4AC$, resulting in $D = 113$.
Because $D > 0$, the equation has two real roots.
Final answer: $\boxed{\text{The discriminant is greater than zero, so there are two real roots.}}$

Explanation

Rearranging to Standard Form

We are given the equation $6x^2 - 8 = 4x^2 + 7x$. To analyze the roots of this equation, we first need to express it in the standard quadratic form $Ax^2 + Bx + C = 0$. We subtract $4x^2$ and $7x$ from both sides of the equation:
\[6x^2 - 4x^2 - 7x - 8 = 0\]
Simplifying the coefficients, we obtain:
\[2x^2 - 7x - 8 = 0\]
This equation is now in standard quadratic form with coefficients $A = 2$, $B = -7$, and $C = -8$.

Calculating the Discriminant

The nature of the roots of a quadratic equation $Ax^2 + Bx + C = 0$ is determined by its discriminant, which is calculated using the formula $D = B^2 - 4AC$. Substituting our coefficients $A = 2$, $B = -7$, and $C = -8$ into this formula, we get:
\[D = (-7)^2 - 4(2)(-8)\]
Calculating the values:
\[D = 49 - (-64)\]
\[D = 49 + 64\]
\[D = 113\]
The discriminant of the equation is $113$.

Determining the Nature of the Roots

Since the discriminant $D = 113$ is greater than zero, the quadratic formula tells us that there are two distinct real roots. Therefore, the statement "The discriminant is greater than zero, so there are two real roots" correctly describes the nature of the roots for the given quadratic equation.

Examples

In engineering, discriminant analysis is used to determine the stability of a system. For instance, if you are designing a suspension system for a car, the equation governing the motion involves a quadratic characteristic equation. If the discriminant is positive, the system is overdamped (returning to equilibrium without oscillation); if it is zero, it is critically damped; and if it is negative, it is underdamped (oscillating). By checking the sign of the discriminant, engineers can ensure the ride is comfortable and safe.