Algebra Examples

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

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

Step 1

Rewrite the equation as

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

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

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

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

Step 2

Subtract

y

$y$ from both sides of the equation.

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

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

Step 3

Use the quadratic formula to find the solutions.

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

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

Step 4

Substitute the values

a = a

$a = a$ ,

b = b

$b = b$ , and

c = c - y

$c = c - y$ into the quadratic formula and solve for

x

$x$ .

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

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

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

Step 5.2

Rewrite using the commutative property of multiplication.

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

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

Step 5.3

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

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

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

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

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

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}

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

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

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