Pre-Algebra Examples

$x^{2} - 10 x + 25 = - 28$

Step 1

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 1.1

Add $28$ to both sides of the equation.

$x^{2} - 10 x + 25 + 28 = 0$

Step 1.2

Add $25$ and $28$ .

$x^{2} - 10 x + 53 = 0$

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 $c = 53$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 53)}}{2 \cdot 1}$

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 $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 53}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 53$ .

Tap for more steps...

Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 53}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $53$ .

$x = \frac{10 \pm \sqrt{100 - 212}}{2 \cdot 1}$

Step 4.1.3

Subtract $212$ from $100$ .

$x = \frac{10 \pm \sqrt{- 112}}{2 \cdot 1}$

Step 4.1.4

Rewrite $- 112$ as $- 1 (112)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 112}}{2 \cdot 1}$

Step 4.1.5

Rewrite $\sqrt{- 1 (112)}$ as $\sqrt{- 1} \cdot \sqrt{112}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{112}}{2 \cdot 1}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{112}}{2 \cdot 1}$

Step 4.1.7

Rewrite $112$ as $4^{2} \cdot 7$ .

Tap for more steps...

Step 4.1.7.1

Factor $16$ out of $112$ .

$x = \frac{10 \pm i \cdot \sqrt{16 (7)}}{2 \cdot 1}$

Step 4.1.7.2

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

$x = \frac{10 \pm i \cdot \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (4 \sqrt{7})}{2 \cdot 1}$

Step 4.1.9

Move $4$ to the left of $i$ .

$x = \frac{10 \pm 4 i \sqrt{7}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{10 \pm 4 i \sqrt{7}}{2}$

Step 4.3

Simplify $\frac{10 \pm 4 i \sqrt{7}}{2}$ .

$x = 5 \pm 2 i \sqrt{7}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

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

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 53}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 53$ .

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 53}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $53$ .

$x = \frac{10 \pm \sqrt{100 - 212}}{2 \cdot 1}$

Step 5.1.3

Subtract $212$ from $100$ .

$x = \frac{10 \pm \sqrt{- 112}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 112$ as $- 1 (112)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 112}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (112)}$ as $\sqrt{- 1} \cdot \sqrt{112}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{112}}{2 \cdot 1}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{112}}{2 \cdot 1}$

Step 5.1.7

Rewrite $112$ as $4^{2} \cdot 7$ .

Tap for more steps...

Step 5.1.7.1

Factor $16$ out of $112$ .

$x = \frac{10 \pm i \cdot \sqrt{16 (7)}}{2 \cdot 1}$

Step 5.1.7.2

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

$x = \frac{10 \pm i \cdot \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (4 \sqrt{7})}{2 \cdot 1}$

Step 5.1.9

Move $4$ to the left of $i$ .

$x = \frac{10 \pm 4 i \sqrt{7}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{10 \pm 4 i \sqrt{7}}{2}$

Step 5.3

Simplify $\frac{10 \pm 4 i \sqrt{7}}{2}$ .

$x = 5 \pm 2 i \sqrt{7}$

Step 5.4

Change the $\pm$ to $+$ .

$x = 5 + 2 i \sqrt{7}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

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

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 53}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 53$ .

Tap for more steps...

Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 53}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $53$ .

$x = \frac{10 \pm \sqrt{100 - 212}}{2 \cdot 1}$

Step 6.1.3

Subtract $212$ from $100$ .

$x = \frac{10 \pm \sqrt{- 112}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 112$ as $- 1 (112)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 112}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (112)}$ as $\sqrt{- 1} \cdot \sqrt{112}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{112}}{2 \cdot 1}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{10 \pm i \cdot \sqrt{112}}{2 \cdot 1}$

Step 6.1.7

Rewrite $112$ as $4^{2} \cdot 7$ .

Tap for more steps...

Step 6.1.7.1

Factor $16$ out of $112$ .

$x = \frac{10 \pm i \cdot \sqrt{16 (7)}}{2 \cdot 1}$

Step 6.1.7.2

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

$x = \frac{10 \pm i \cdot \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (4 \sqrt{7})}{2 \cdot 1}$

Step 6.1.9

Move $4$ to the left of $i$ .

$x = \frac{10 \pm 4 i \sqrt{7}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{10 \pm 4 i \sqrt{7}}{2}$

Step 6.3

Simplify $\frac{10 \pm 4 i \sqrt{7}}{2}$ .

$x = 5 \pm 2 i \sqrt{7}$

Step 6.4

Change the $\pm$ to $-$ .

$x = 5 - 2 i \sqrt{7}$

Step 7

The final answer is the combination of both solutions.

$x = 5 + 2 i \sqrt{7}, 5 - 2 i \sqrt{7}$