Question

Four times the square of a number is 21 more than eight times the number.

What is the number?
Worth fifty points!

Answer 1

Number be n

• 4n²=21+8n
• 4n²-8n-21=0
• 4n²-14n+6n-21=0
• 2n(2n-7)+3(2n-7)=0
• (2n+3(2n-7)=0

n=-3/2 and 7/2

Answer 2

`x=-(3)/(2), x=(7)/(2)`

Explanation:

Let the unknown number = x

`\implies 4x^2=21+8x`

`\implies 4x^2-8x-21=0`

Factor the equation

Find factors of -84 that sum to -8: -14 and 6

Rewrite -8x as the sum of these 2 numbers:

`\implies 4x^2-14x+6x-21=0`

Factorize the first two terms and the last two terms separately:

`\implies 2x(2x-7)+3(2x-7)=0`

Factor out the common term
`(2x-7)`:

`\implies (2x+3)(2x-7)=0`

Therefore,

`\implies 2x+3=0 \implies x=-(3)/(2)`

`\implies 2x-7=0 \implies x=(7)/(2)`