Question

1. which equations are quadratic? select all that apply.

□ 6x²−2x+8=0
□ 2x³+4x²−x−9=0
□ (4x−2)²+14=0
□ (9x−3)⁴−2=0

Explanation:

Step 1: Recall the definition of a quadratic equation

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable (usually \( x \)) is 2. The general form is \( ax^2 + bx + c = 0 \), where \( a
eq 0 \).

Step 2: Analyze each equation

• For \( 6x^2 - 2x + 8 = 0 \):

The highest power of \( x \) is 2 (from \( 6x^2 \)), and it is in the form \( ax^2 + bx + c = 0 \) with \( a = 6
eq 0 \). So this is a quadratic equation.

• For \( 2x^3 + 4x^2 - x - 9 = 0 \):

The highest power of \( x \) is 3 (from \( 2x^3 \)), so this is a cubic (degree 3) equation, not quadratic.

• For \( (4x - 2)^2 + 14 = 0 \):

Expand \( (4x - 2)^2 \):
\( (4x - 2)^2 = (4x)^2 - 2(4x)(2) + 2^2 = 16x^2 - 16x + 4 \).
Then the equation becomes \( 16x^2 - 16x + 4 + 14 = 0 \), or \( 16x^2 - 16x + 18 = 0 \).
The highest power of \( x \) is 2, so this is a quadratic equation.

• For \( (9x - 3)^4 - 2 = 0 \):

The highest power of \( x \) is 4 (from \( (9x - 3)^4 \)), so this is a quartic (degree 4) equation, not quadratic.

Answer:

A. \( 6x^2 - 2x + 8 = 0 \)
C. \( (4x - 2)^2 + 14 = 0 \)