Question

Which function is equivalent to f(x) =2x² - 8x + 3?

Answer 1

$g(x) = 2(x-2)^2-5 \text{ is equivalent to } 2x^2-8x+3$

Solution:

Given expression is:

$f(x) = 2x^2-8x+3$

We have to find the equivalent expression

$\mathrm{Write}\:2x^2-8x+3\:\mathrm{in\:the\:form:\:\:}x^2+2ax+a^2$

$f(x) = 2x^2-8x+3$

$\mathrm{Factor\:out\:}2$

$2\left(x^2-4x+\frac{3}{2}\right)$

$\mathrm{Add\:and\:subtract}\:\left(-2\right)^2\:$

$2\left(x^2-4x+\frac{3}{2}+\left(-2\right)^2-\left(-2\right)^2\right)$

Simplify the above equation

$2(x^2-4x + \frac{3}{2} + 4 -4)\\\\2(x^2-4x +4 + \frac{3}{2} - 4)\\\\2(x^2-4x +4 -\frac{5}{2}) ------- eqn 1$

Using the algebraic identity,

$x^2+2ax+a^2=\left(x+a\right)^2$

Therefore,

$x^2-4x+4=\left(x-2\right)^2$

Substitute the above in eqn 1

$2((x-2)^2-\frac{5}{2})$

Simplify the above equation by multiplying 2 with terms inside the bracket

$2(x-2)^2 -5$

Thus the equivalent expression is $g(x) = 2(x-2)^2-5$