Basic Math Examples

$x^{2} + y^{2} + 4 x - 10 y = 7$

Step 1

Subtract $7$ from both sides of the equation.

$x^{2} + y^{2} + 4 x - 10 y - 7 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - 10$ , and $c = x^{2} + 4 x - 7$ into the quadratic formula and solve for $y$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot (x^{2} + 4 x - 7))}}{2 \cdot 1}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot (x^{2} + 4 x - 7)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot (x^{2} + 4 x - 7)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$y = \frac{10 \pm \sqrt{100 - 4 x^{2} - 4 (4 x) - 4 \cdot - 7}}{2 \cdot 1}$

Step 4.1.4

Simplify.

Tap for more steps...

Step 4.1.4.1

Multiply $4$ by $- 4$ .

$y = \frac{10 \pm \sqrt{100 - 4 x^{2} - 16 x - 4 \cdot - 7}}{2 \cdot 1}$

Step 4.1.4.2

Multiply $- 4$ by $- 7$ .

$y = \frac{10 \pm \sqrt{100 - 4 x^{2} - 16 x + 28}}{2 \cdot 1}$

Step 4.1.5

Add $100$ and $28$ .

$y = \frac{10 \pm \sqrt{- 4 x^{2} - 16 x + 128}}{2 \cdot 1}$

Step 4.1.6

Rewrite $- 4 x^{2} - 16 x + 128$ in a factored form.

Tap for more steps...

Step 4.1.6.1

Factor $4$ out of $- 4 x^{2} - 16 x + 128$ .

Tap for more steps...

Step 4.1.6.1.1

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

$y = \frac{10 \pm \sqrt{4 (- x^{2}) - 16 x + 128}}{2 \cdot 1}$

Step 4.1.6.1.2

Factor $4$ out of $- 16 x$ .

$y = \frac{10 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 128}}{2 \cdot 1}$

Step 4.1.6.1.3

Factor $4$ out of $128$ .

$y = \frac{10 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 4 (32)}}{2 \cdot 1}$

Step 4.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (- 4 x)$ .

$y = \frac{10 \pm \sqrt{4 (- x^{2} - 4 x) + 4 (32)}}{2 \cdot 1}$

Step 4.1.6.1.5

Factor $4$ out of $4 (- x^{2} - 4 x) + 4 (32)$ .

$y = \frac{10 \pm \sqrt{4 (- x^{2} - 4 x + 32)}}{2 \cdot 1}$

Step 4.1.6.2

Factor by grouping.

Tap for more steps...

Step 4.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 32 = - 32$ and whose sum is $b = - 4$ .

Tap for more steps...

Step 4.1.6.2.1.1

Factor $- 4$ out of $- 4 x$ .

$y = \frac{10 \pm \sqrt{4 (- x^{2} - 4 x + 32)}}{2 \cdot 1}$

Step 4.1.6.2.1.2

Rewrite $- 4$ as $4$ plus $- 8$

$y = \frac{10 \pm \sqrt{4 (- x^{2} + (4 - 8) x + 32)}}{2 \cdot 1}$

Step 4.1.6.2.1.3

Apply the distributive property.

$y = \frac{10 \pm \sqrt{4 (- x^{2} + 4 x - 8 x + 32)}}{2 \cdot 1}$

Step 4.1.6.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{10 \pm \sqrt{4 ((- x^{2} + 4 x) - 8 x + 32)}}{2 \cdot 1}$

Step 4.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{10 \pm \sqrt{4 (x (- x + 4) + 8 (- x + 4))}}{2 \cdot 1}$

Step 4.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 4$ .

$y = \frac{10 \pm \sqrt{4 ((- x + 4) (x + 8))}}{2 \cdot 1}$

$y = \frac{10 \pm \sqrt{4 (- x + 4) (x + 8)}}{2 \cdot 1}$

Step 4.1.7

Rewrite $4 (- x + 4) (x + 8)$ as $2^{2} ((- x + 2^{2}) (x + 8))$ .

Tap for more steps...

Step 4.1.7.1

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

$y = \frac{10 \pm \sqrt{2^{2} (- x + 4) (x + 8)}}{2 \cdot 1}$

Step 4.1.7.2

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

$y = \frac{10 \pm \sqrt{2^{2} (- x + 2^{2}) (x + 8)}}{2 \cdot 1}$

Step 4.1.7.3

Add parentheses.

$y = \frac{10 \pm \sqrt{2^{2} ((- x + 2^{2}) (x + 8))}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{(- x + 2^{2}) (x + 8)}}{2 \cdot 1}$

Step 4.1.9

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

$y = \frac{10 \pm 2 \sqrt{(- x + 4) (x + 8)}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{10 \pm 2 \sqrt{(- x + 4) (x + 8)}}{2}$

Step 4.3

Simplify $\frac{10 \pm 2 \sqrt{(- x + 4) (x + 8)}}{2}$ .

$y = 5 \pm \sqrt{(- x + 4) (x + 8)}$

Step 5

The final answer is the combination of both solutions.

$y = 5 + \sqrt{(- x + 4) (x + 8)}$

$y = 5 - \sqrt{(- x + 4) (x + 8)}$