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:

⚡ Using what you learned: quadratic formula and its applications

Step 1: Write in standard form

Rearrange the given equation \( 6x^2 - 8 = 4x^2 + 7x \) into the standard quadratic form \( ax^2 + bx + c = 0 \):

$$ (6x^2 - 4x^2) - 7x - 8 = 0 $$

$$ 2x^2 - 7x - 8 = 0 $$

Identify the coefficients:

• \( a = 2 \)
• \( b = -7 \)
• \( c = -8 \)

Step 2: Calculate the discriminant

The discriminant formula is \( D = b^2 - 4ac \):

$$ D = (-7)^2 - 4(2)(-8) $$

$$ D = 49 - (-64) $$

$$ D = 49 + 64 = 113 $$

Step 3: Determine the nature of the roots

Since the discriminant \( D = 113 \) is greater than zero (\( D > 0 \)), the quadratic equation has two distinct real roots.

Answer:

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