Basic Math Examples

$4 y^{2} + 4 y - 24 x + 35 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 4$ , $b = 4$ , and $c = - 24 x + 35$ into the quadratic formula and solve for $y$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (4 \cdot (- 24 x + 35))}}{2 \cdot 4}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Factor $4$ out of $4^{2} - 4 \cdot 4 \cdot (- 24 x + 35)$ .

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Step 3.1.1.1

Factor $4$ out of $4^{2}$ .

$y = \frac{- 4 \pm \sqrt{4 \cdot 4 - 4 \cdot 4 \cdot (- 24 x + 35)}}{2 \cdot 4}$

Step 3.1.1.2

Factor $4$ out of $- 4 \cdot 4 \cdot (- 24 x + 35)$ .

$y = \frac{- 4 \pm \sqrt{4 \cdot 4 + 4 (- 1 \cdot 4 \cdot (- 24 x + 35))}}{2 \cdot 4}$

Step 3.1.1.3

Factor $4$ out of $4 \cdot 4 + 4 (- 1 \cdot 4 \cdot (- 24 x + 35))$ .

$y = \frac{- 4 \pm \sqrt{4 (4 - 1 \cdot 4 \cdot (- 24 x + 35))}}{2 \cdot 4}$

Step 3.1.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 \pm \sqrt{4 (4 - 4 \cdot (- 24 x + 35))}}{2 \cdot 4}$

Step 3.1.3

Apply the distributive property.

$y = \frac{- 4 \pm \sqrt{4 (4 - 4 (- 24 x) - 4 \cdot 35)}}{2 \cdot 4}$

Step 3.1.4

Multiply $- 24$ by $- 4$ .

$y = \frac{- 4 \pm \sqrt{4 (4 + 96 x - 4 \cdot 35)}}{2 \cdot 4}$

Step 3.1.5

Multiply $- 4$ by $35$ .

$y = \frac{- 4 \pm \sqrt{4 (4 + 96 x - 140)}}{2 \cdot 4}$

Step 3.1.6

Subtract $140$ from $4$ .

$y = \frac{- 4 \pm \sqrt{4 (96 x - 136)}}{2 \cdot 4}$

Step 3.1.7

Factor $8$ out of $96 x - 136$ .

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Step 3.1.7.1

Factor $8$ out of $96 x$ .

$y = \frac{- 4 \pm \sqrt{4 (8 (12 x) - 136)}}{2 \cdot 4}$

Step 3.1.7.2

Factor $8$ out of $- 136$ .

$y = \frac{- 4 \pm \sqrt{4 (8 (12 x) + 8 (- 17))}}{2 \cdot 4}$

Step 3.1.7.3

Factor $8$ out of $8 (12 x) + 8 (- 17)$ .

$y = \frac{- 4 \pm \sqrt{4 (8 (12 x - 17))}}{2 \cdot 4}$

$y = \frac{- 4 \pm \sqrt{4 \cdot (8 (12 x - 17))}}{2 \cdot 4}$

Step 3.1.8

Multiply $4$ by $8$ .

$y = \frac{- 4 \pm \sqrt{32 (12 x - 17)}}{2 \cdot 4}$

Step 3.1.9

Rewrite $32 (12 x - 17)$ as ${(2^{2})}^{2} \cdot (2 (12 x - 17))$ .

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Step 3.1.9.1

Factor $16$ out of $32$ .

$y = \frac{- 4 \pm \sqrt{16 (2) (12 x - 17)}}{2 \cdot 4}$

Step 3.1.9.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 4 \pm \sqrt{4^{2} \cdot (2 (12 x - 17))}}{2 \cdot 4}$

Step 3.1.9.3

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 4 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (12 x - 17))}}{2 \cdot 4}$

Step 3.1.9.4

Add parentheses.

$y = \frac{- 4 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (12 x - 17))}}{2 \cdot 4}$

Step 3.1.10

Pull terms out from under the radical.

$y = \frac{- 4 \pm 2^{2} \sqrt{2 (12 x - 17)}}{2 \cdot 4}$

Step 3.1.11

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

$y = \frac{- 4 \pm 4 \sqrt{2 (12 x - 17)}}{2 \cdot 4}$

Step 3.2

Multiply $2$ by $4$ .

$y = \frac{- 4 \pm 4 \sqrt{2 (12 x - 17)}}{8}$

Step 3.3

Simplify $\frac{- 4 \pm 4 \sqrt{2 (12 x - 17)}}{8}$ .

$y = \frac{- 1 \pm \sqrt{2 (12 x - 17)}}{2}$

Step 4

The final answer is the combination of both solutions.

$y = - \frac{1 - \sqrt{2 (12 x - 17)}}{2}$

$y = - \frac{1 + \sqrt{2 (12 x - 17)}}{2}$