Find the Vertex y=5x^2+2x-1

$y = 5 x^{2} + 2 x - 1$

Step 1

Rewrite the equation in vertex form.

Complete the square for $5 x^{2} + 2 x - 1$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 5$

$b = 2$

$c = - 1$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot 5}$

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 \cdot 5}$

Rewrite the expression.

$d = \frac{1}{5}$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 1 - \frac{2^{2}}{4 \cdot 5}$

Simplify the right side.

Simplify each term.

Raise $2$ to the power of $2$ .

$e = - 1 - \frac{4}{4 \cdot 5}$

Multiply $4$ by $5$ .

$e = - 1 - \frac{4}{20}$

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $4$ .

$e = - 1 - \frac{4 (1)}{20}$

Cancel the common factors.

Factor $4$ out of $20$ .

$e = - 1 - \frac{4 \cdot 1}{4 \cdot 5}$

Cancel the common factor.

$e = - 1 - \frac{4 \cdot 1}{4 \cdot 5}$

Rewrite the expression.

$e = - 1 - \frac{1}{5}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$e = - 1 \cdot \frac{5}{5} - \frac{1}{5}$

Combine $- 1$ and $\frac{5}{5}$ .

$e = \frac{- 1 \cdot 5}{5} - \frac{1}{5}$

Combine the numerators over the common denominator.

$e = \frac{- 1 \cdot 5 - 1}{5}$

Simplify the numerator.

Multiply $- 1$ by $5$ .

$e = \frac{- 5 - 1}{5}$

Subtract $1$ from $- 5$ .

$e = \frac{- 6}{5}$

Move the negative in front of the fraction.

$e = - \frac{6}{5}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x + \frac{1}{5})}^{2} - \frac{6}{5}$ .

$5 {(x + \frac{1}{5})}^{2} - \frac{6}{5}$

Set $y$ equal to the new right side.

$y = 5 {(x + \frac{1}{5})}^{2} - \frac{6}{5}$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 5$

$h = - \frac{1}{5}$

$k = - \frac{6}{5}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{1}{5}, - \frac{6}{5})$

Step 4

Find the Vertex y=5x^2+2x-1

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