Algebra Examples

$x + 10 = 3 {(x - 1)}^{2}$

Step 1

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

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $3 {(x - 1)}^{2}$ .

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Step 1.1.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x + 10 = 3 ((x - 1) (x - 1))$

Step 1.1.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.1.1.2.1

Apply the distributive property.

$x + 10 = 3 (x (x - 1) - 1 (x - 1))$

Step 1.1.1.2.2

Apply the distributive property.

$x + 10 = 3 (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Step 1.1.1.2.3

Apply the distributive property.

$x + 10 = 3 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.3.1

Simplify each term.

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Step 1.1.1.3.1.1

Multiply $x$ by $x$ .

$x + 10 = 3 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 1.1.1.3.1.2

Move $- 1$ to the left of $x$ .

$x + 10 = 3 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Step 1.1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$x + 10 = 3 (x^{2} - x - 1 x - 1 \cdot - 1)$

Step 1.1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$x + 10 = 3 (x^{2} - x - x - 1 \cdot - 1)$

Step 1.1.1.3.1.5

Multiply $- 1$ by $- 1$ .

$x + 10 = 3 (x^{2} - x - x + 1)$

Step 1.1.1.3.2

Subtract $x$ from $- x$ .

$x + 10 = 3 (x^{2} - 2 x + 1)$

Step 1.1.1.4

Apply the distributive property.

$x + 10 = 3 x^{2} + 3 (- 2 x) + 3 \cdot 1$

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Multiply $- 2$ by $3$ .

$x + 10 = 3 x^{2} - 6 x + 3 \cdot 1$

Step 1.1.1.5.2

Multiply $3$ by $1$ .

$x + 10 = 3 x^{2} - 6 x + 3$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract $3 x^{2}$ from both sides of the equation.

$x + 10 - 3 x^{2} = - 6 x + 3$

Step 1.2.2

Add $6 x$ to both sides of the equation.

$x + 10 - 3 x^{2} + 6 x = 3$

Step 1.2.3

Subtract $3$ from both sides of the equation.

$x + 10 - 3 x^{2} + 6 x - 3 = 0$

Step 1.3

Simplify $x + 10 - 3 x^{2} + 6 x - 3$ .

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Step 1.3.1

Add $x$ and $6 x$ .

$7 x + 10 - 3 x^{2} - 3 = 0$

Step 1.3.2

Subtract $3$ from $10$ .

$7 x - 3 x^{2} + 7 = 0$

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 $a = - 3$ , $b = 7$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (- 3 \cdot 7)}}{2 \cdot - 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 3 \cdot 7}}{2 \cdot - 3}$

Step 4.1.2

Multiply $- 4 \cdot - 3 \cdot 7$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 7 \pm \sqrt{49 + 12 \cdot 7}}{2 \cdot - 3}$

Step 4.1.2.2

Multiply $12$ by $7$ .

$x = \frac{- 7 \pm \sqrt{49 + 84}}{2 \cdot - 3}$

Step 4.1.3

Add $49$ and $84$ .

$x = \frac{- 7 \pm \sqrt{133}}{2 \cdot - 3}$

Step 4.2

Multiply $2$ by $- 3$ .

$x = \frac{- 7 \pm \sqrt{133}}{- 6}$

Step 4.3

Simplify $\frac{- 7 \pm \sqrt{133}}{- 6}$ .

$x = \frac{7 \pm \sqrt{133}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 \pm \sqrt{133}}{6}$

Decimal Form:

$x = 3.08876043 \dots, - 0.75542709 \dots$