Precalculus Examples Systems of Equations

Find the Equation with Real Coefficients

$x = 9$ , $x = - 9$ , $x = 2$

Step 1

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

$(x - 9) (x - (- 9)) (x - 2) = 0$

Step 2

Simplify.

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Step 2.1

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

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Step 2.1.1

Apply the distributive property.

$(x (x + 9) - 9 (x + 9)) (x - 2) = 0$

Step 2.1.2

Apply the distributive property.

$(x \cdot x + x \cdot 9 - 9 (x + 9)) (x - 2) = 0$

Step 2.1.3

Apply the distributive property.

$(x \cdot x + x \cdot 9 - 9 x - 9 \cdot 9) (x - 2) = 0$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine the opposite terms in $x \cdot x + x \cdot 9 - 9 x - 9 \cdot 9$ .

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Step 2.2.1.1

Reorder the factors in the terms $x \cdot 9$ and $- 9 x$ .

$(x \cdot x + 9 x - 9 x - 9 \cdot 9) (x - 2) = 0$

Step 2.2.1.2

Subtract $9 x$ from $9 x$ .

$(x \cdot x + 0 - 9 \cdot 9) (x - 2) = 0$

Step 2.2.1.3

Add $x \cdot x$ and $0$ .

$(x \cdot x - 9 \cdot 9) (x - 2) = 0$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.2.2

Multiply $- 9$ by $9$ .

$(x^{2} - 81) (x - 2) = 0$

Step 2.3

Expand $(x^{2} - 81) (x - 2)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$x^{2} (x - 2) - 81 (x - 2) = 0$

Step 2.3.2

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 2 - 81 (x - 2) = 0$

Step 2.3.3

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 2 - 81 x - 81 \cdot - 2 = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.4.1.1

Multiply $x^{2}$ by $x$ .

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Step 2.4.1.1.1

Raise $x$ to the power of $1$ .

$x^{2} x + x^{2} \cdot - 2 - 81 x - 81 \cdot - 2 = 0$

Step 2.4.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1} + x^{2} \cdot - 2 - 81 x - 81 \cdot - 2 = 0$

Step 2.4.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot - 2 - 81 x - 81 \cdot - 2 = 0$

Step 2.4.2

Move $- 2$ to the left of $x^{2}$ .

$x^{3} - 2 \cdot x^{2} - 81 x - 81 \cdot - 2 = 0$

Step 2.4.3

Multiply $- 81$ by $- 2$ .

$x^{3} - 2 x^{2} - 81 x + 162 = 0$

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