Solve for x y=ax^2+bx+c

$y = a x^{2} + b x + c$

Step 1

Rewrite the equation as $a x^{2} + b x + c = y$ .

$a x^{2} + b x + c = y$

Step 2

Subtract $y$ from both sides of the equation.

$a x^{2} + b x + c - y = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = a$ , $b = b$ , and $c = c - y$ into the quadratic formula and solve for $x$ .

$\frac{- b \pm \sqrt{b^{2} - 4 \cdot (a \cdot (c - y))}}{2 a}$

Step 5

Simplify the numerator.

Apply the distributive property.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c - 4 a (- y)}}{2 a}$

Rewrite using the commutative property of multiplication.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c - 4 \cdot (- 1 a y)}}{2 a}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c + 4 a y}}{2 a}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{b - \sqrt{b^{2} - 4 a c + 4 a y}}{2 a}$

$x = - \frac{b + \sqrt{b^{2} - 4 a c + 4 a y}}{2 a}$