Precalculus Examples

${(5 x + 9)}^{2} = 10$

Step 1

Subtract $10$ from both sides of the equation.

${(5 x + 9)}^{2} - 10 = 0$

Step 2

Simplify ${(5 x + 9)}^{2} - 10$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(5 x + 9)}^{2}$ as $(5 x + 9) (5 x + 9)$ .

$(5 x + 9) (5 x + 9) - 10 = 0$

Step 2.1.2

Expand $(5 x + 9) (5 x + 9)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$5 x (5 x + 9) + 9 (5 x + 9) - 10 = 0$

Step 2.1.2.2

Apply the distributive property.

$5 x (5 x) + 5 x \cdot 9 + 9 (5 x + 9) - 10 = 0$

Step 2.1.2.3

Apply the distributive property.

$5 x (5 x) + 5 x \cdot 9 + 9 (5 x) + 9 \cdot 9 - 10 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

$5 \cdot 5 x \cdot x + 5 x \cdot 9 + 9 (5 x) + 9 \cdot 9 - 10 = 0$

Step 2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.3.1.2.1

Move $x$ .

$5 \cdot 5 (x \cdot x) + 5 x \cdot 9 + 9 (5 x) + 9 \cdot 9 - 10 = 0$

Step 2.1.3.1.2.2

Multiply $x$ by $x$ .

$5 \cdot 5 x^{2} + 5 x \cdot 9 + 9 (5 x) + 9 \cdot 9 - 10 = 0$

Step 2.1.3.1.3

Multiply $5$ by $5$ .

$25 x^{2} + 5 x \cdot 9 + 9 (5 x) + 9 \cdot 9 - 10 = 0$

Step 2.1.3.1.4

Multiply $9$ by $5$ .

$25 x^{2} + 45 x + 9 (5 x) + 9 \cdot 9 - 10 = 0$

Step 2.1.3.1.5

Multiply $5$ by $9$ .

$25 x^{2} + 45 x + 45 x + 9 \cdot 9 - 10 = 0$

Step 2.1.3.1.6

Multiply $9$ by $9$ .

$25 x^{2} + 45 x + 45 x + 81 - 10 = 0$

Step 2.1.3.2

Add $45 x$ and $45 x$ .

$25 x^{2} + 90 x + 81 - 10 = 0$

Step 2.2

Subtract $10$ from $81$ .

$25 x^{2} + 90 x + 71 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 25$ , $b = 90$ , and $c = 71$ into the quadratic formula and solve for $x$ .

$\frac{- 90 \pm \sqrt{90^{2} - 4 \cdot (25 \cdot 71)}}{2 \cdot 25}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $90$ to the power of $2$ .

$x = \frac{- 90 \pm \sqrt{8100 - 4 \cdot 25 \cdot 71}}{2 \cdot 25}$

Step 5.1.2

Multiply $- 4 \cdot 25 \cdot 71$ .

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Step 5.1.2.1

Multiply $- 4$ by $25$ .

$x = \frac{- 90 \pm \sqrt{8100 - 100 \cdot 71}}{2 \cdot 25}$

Step 5.1.2.2

Multiply $- 100$ by $71$ .

$x = \frac{- 90 \pm \sqrt{8100 - 7100}}{2 \cdot 25}$

Step 5.1.3

Subtract $7100$ from $8100$ .

$x = \frac{- 90 \pm \sqrt{1000}}{2 \cdot 25}$

Step 5.1.4

Rewrite $1000$ as $10^{2} \cdot 10$ .

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Step 5.1.4.1

Factor $100$ out of $1000$ .

$x = \frac{- 90 \pm \sqrt{100 (10)}}{2 \cdot 25}$

Step 5.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 90 \pm \sqrt{10^{2} \cdot 10}}{2 \cdot 25}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 90 \pm 10 \sqrt{10}}{2 \cdot 25}$

Step 5.2

Multiply $2$ by $25$ .

$x = \frac{- 90 \pm 10 \sqrt{10}}{50}$

Step 5.3

Simplify $\frac{- 90 \pm 10 \sqrt{10}}{50}$ .

$x = \frac{- 9 \pm \sqrt{10}}{5}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $90$ to the power of $2$ .

$x = \frac{- 90 \pm \sqrt{8100 - 4 \cdot 25 \cdot 71}}{2 \cdot 25}$

Step 6.1.2

Multiply $- 4 \cdot 25 \cdot 71$ .

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Step 6.1.2.1

Multiply $- 4$ by $25$ .

$x = \frac{- 90 \pm \sqrt{8100 - 100 \cdot 71}}{2 \cdot 25}$

Step 6.1.2.2

Multiply $- 100$ by $71$ .

$x = \frac{- 90 \pm \sqrt{8100 - 7100}}{2 \cdot 25}$

Step 6.1.3

Subtract $7100$ from $8100$ .

$x = \frac{- 90 \pm \sqrt{1000}}{2 \cdot 25}$

Step 6.1.4

Rewrite $1000$ as $10^{2} \cdot 10$ .

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Step 6.1.4.1

Factor $100$ out of $1000$ .

$x = \frac{- 90 \pm \sqrt{100 (10)}}{2 \cdot 25}$

Step 6.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 90 \pm \sqrt{10^{2} \cdot 10}}{2 \cdot 25}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 90 \pm 10 \sqrt{10}}{2 \cdot 25}$

Step 6.2

Multiply $2$ by $25$ .

$x = \frac{- 90 \pm 10 \sqrt{10}}{50}$

Step 6.3

Simplify $\frac{- 90 \pm 10 \sqrt{10}}{50}$ .

$x = \frac{- 9 \pm \sqrt{10}}{5}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{- 9 + \sqrt{10}}{5}$

Step 6.5

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 + \sqrt{10}}{5}$

Step 6.6

Factor $- 1$ out of $\sqrt{10}$ .

$x = \frac{- 1 \cdot 9 - 1 (- \sqrt{10})}{5}$

Step 6.7

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{10})$ .

$x = \frac{- 1 (9 - \sqrt{10})}{5}$

Step 6.8

Move the negative in front of the fraction.

$x = - \frac{9 - \sqrt{10}}{5}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $90$ to the power of $2$ .

$x = \frac{- 90 \pm \sqrt{8100 - 4 \cdot 25 \cdot 71}}{2 \cdot 25}$

Step 7.1.2

Multiply $- 4 \cdot 25 \cdot 71$ .

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Step 7.1.2.1

Multiply $- 4$ by $25$ .

$x = \frac{- 90 \pm \sqrt{8100 - 100 \cdot 71}}{2 \cdot 25}$

Step 7.1.2.2

Multiply $- 100$ by $71$ .

$x = \frac{- 90 \pm \sqrt{8100 - 7100}}{2 \cdot 25}$

Step 7.1.3

Subtract $7100$ from $8100$ .

$x = \frac{- 90 \pm \sqrt{1000}}{2 \cdot 25}$

Step 7.1.4

Rewrite $1000$ as $10^{2} \cdot 10$ .

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Step 7.1.4.1

Factor $100$ out of $1000$ .

$x = \frac{- 90 \pm \sqrt{100 (10)}}{2 \cdot 25}$

Step 7.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 90 \pm \sqrt{10^{2} \cdot 10}}{2 \cdot 25}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 90 \pm 10 \sqrt{10}}{2 \cdot 25}$

Step 7.2

Multiply $2$ by $25$ .

$x = \frac{- 90 \pm 10 \sqrt{10}}{50}$

Step 7.3

Simplify $\frac{- 90 \pm 10 \sqrt{10}}{50}$ .

$x = \frac{- 9 \pm \sqrt{10}}{5}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{- 9 - \sqrt{10}}{5}$

Step 7.5

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 - \sqrt{10}}{5}$

Step 7.6

Factor $- 1$ out of $- \sqrt{10}$ .

$x = \frac{- 1 \cdot 9 - (\sqrt{10})}{5}$

Step 7.7

Factor $- 1$ out of $- 1 (9) - (\sqrt{10})$ .

$x = \frac{- 1 (9 + \sqrt{10})}{5}$

Step 7.8

Move the negative in front of the fraction.

$x = - \frac{9 + \sqrt{10}}{5}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{9 - \sqrt{10}}{5}, - \frac{9 + \sqrt{10}}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 - \sqrt{10}}{5}, - \frac{9 + \sqrt{10}}{5}$

Decimal Form:

$x = - 1.16754446 \dots, - 2.43245553 \dots$