Question

convert the equation y = 5x² - 40x + 67 into vertex form.
y = 5(x + 4)²+13
y = 5(x - 13)²+4
y = 5(x + 4)²-13
y = 5(x - 4)²-13

Explanation:

Step1: Recall vertex - form of quadratic equation

The vertex - form of a quadratic equation is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. Given the quadratic equation $y = 5x^{2}-40x + 67$.
First, factor out the coefficient of $x^{2}$ from the first two terms: $y=5(x^{2}-8x)+67$.

Step2: Complete the square inside the parentheses

For the quadratic expression $x^{2}-8x$, we know that $(x - m)^2=x^{2}-2mx + m^{2}$. In $x^{2}-8x$, $2m = 8$, so $m = 4$ and $x^{2}-8x=(x - 4)^{2}-16$.
Substitute $x^{2}-8x=(x - 4)^{2}-16$ into the equation: $y = 5((x - 4)^{2}-16)+67$.

Step3: Expand and simplify

$y=5(x - 4)^{2}-80 + 67$.
$y=5(x - 4)^{2}-13$.

Answer:

$y = 5(x - 4)^{2}-13$