Find the Equation with a Point and y-intercept y=8x^2+bx+c (5,-5)

$y = 8 x^{2} + b x + c$ $(5, - 5)$

Step 1

Find the value of $m$ using the formula for the equation of a line.

$y = m x + b$

Step 2

Substitute the value of $m$ into the equation.

$y = m x + 8 x^{2} + b x + c$

Step 3

Substitute the value of $x$ into the equation.

$y = m (5) + 8 {(5)}^{2} + b (5) + c$

Step 4

Substitute the value of $y$ into the equation.

$- 5 = m (5) + 8 {(5)}^{2} + b (5) + c$

Step 5

Find the value of $m$ .

Rewrite the equation as $m (5) + 8 {(5)}^{2} + b (5) + c = - 5$ .

$m (5) + 8 {(5)}^{2} + b (5) + c = - 5$

Simplify each term.

Move $5$ to the left of $m$ .

$5 m + 8 {(5)}^{2} + b (5) + c = - 5$

Raise $5$ to the power of $2$ .

$5 m + 8 \cdot 25 + b (5) + c = - 5$

Multiply $8$ by $25$ .

$5 m + 200 + b (5) + c = - 5$

Move $5$ to the left of $b$ .

$5 m + 200 + 5 b + c = - 5$

Move all terms not containing $m$ to the right side of the equation.

Subtract $200$ from both sides of the equation.

$5 m + 5 b + c = - 5 - 200$

Subtract $5 b$ from both sides of the equation.

$5 m + c = - 5 - 200 - 5 b$

Subtract $c$ from both sides of the equation.

$5 m = - 5 - 200 - 5 b - c$

Subtract $200$ from $- 5$ .

$5 m = - 205 - 5 b - c$

Divide each term in $5 m = - 205 - 5 b - c$ by $5$ and simplify.

Divide each term in $5 m = - 205 - 5 b - c$ by $5$ .

$\frac{5 m}{5} = \frac{- 205}{5} + \frac{- 5 b}{5} + \frac{- c}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 m}{5} = \frac{- 205}{5} + \frac{- 5 b}{5} + \frac{- c}{5}$

Divide $m$ by $1$ .

$m = \frac{- 205}{5} + \frac{- 5 b}{5} + \frac{- c}{5}$

Simplify the right side.

Simplify each term.

Divide $- 205$ by $5$ .

$m = - 41 + \frac{- 5 b}{5} + \frac{- c}{5}$

Cancel the common factor of $- 5$ and $5$ .

Factor $5$ out of $- 5 b$ .

$m = - 41 + \frac{5 (- b)}{5} + \frac{- c}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$m = - 41 + \frac{5 (- b)}{5 (1)} + \frac{- c}{5}$

Cancel the common factor.

$m = - 41 + \frac{5 (- b)}{5 \cdot 1} + \frac{- c}{5}$

Rewrite the expression.

$m = - 41 + \frac{- b}{1} + \frac{- c}{5}$

Divide $- b$ by $1$ .

$m = - 41 - b + \frac{- c}{5}$

Move the negative in front of the fraction.

$m = - 41 - b - \frac{c}{5}$

Step 6

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = (- 41 - b - \frac{1}{5} c) x + 8 x^{2} + b x + c$

Step 7

Find the Equation with a Point and y-intercept y=8x^2+bx+c (5,-5)

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