Pre-Algebra Examples

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

Step 1

Simplify ${(x^{2} - 7)}^{2} - 3 (x^{2} - 7) + 2$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x^{2} - 7)}^{2}$ as $(x^{2} - 7) (x^{2} - 7)$ .

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

Step 1.1.2

Expand $(x^{2} - 7) (x^{2} - 7)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot - 7 - 7 (x^{2} - 7) - 3 (x^{2} - 7) + 2 = 0$

Step 1.1.2.3

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot - 7 - 7 x^{2} - 7 \cdot - 7 - 3 (x^{2} - 7) + 2 = 0$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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Step 1.1.3.1.1.1

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

$x^{2 + 2} + x^{2} \cdot - 7 - 7 x^{2} - 7 \cdot - 7 - 3 (x^{2} - 7) + 2 = 0$

Step 1.1.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot - 7 - 7 x^{2} - 7 \cdot - 7 - 3 (x^{2} - 7) + 2 = 0$

Step 1.1.3.1.2

Move $- 7$ to the left of $x^{2}$ .

$x^{4} - 7 \cdot x^{2} - 7 x^{2} - 7 \cdot - 7 - 3 (x^{2} - 7) + 2 = 0$

Step 1.1.3.1.3

Multiply $- 7$ by $- 7$ .

$x^{4} - 7 x^{2} - 7 x^{2} + 49 - 3 (x^{2} - 7) + 2 = 0$

Step 1.1.3.2

Subtract $7 x^{2}$ from $- 7 x^{2}$ .

$x^{4} - 14 x^{2} + 49 - 3 (x^{2} - 7) + 2 = 0$

Step 1.1.4

Apply the distributive property.

$x^{4} - 14 x^{2} + 49 - 3 x^{2} - 3 \cdot - 7 + 2 = 0$

Step 1.1.5

Multiply $- 3$ by $- 7$ .

$x^{4} - 14 x^{2} + 49 - 3 x^{2} + 21 + 2 = 0$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Subtract $3 x^{2}$ from $- 14 x^{2}$ .

$x^{4} - 17 x^{2} + 49 + 21 + 2 = 0$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $49$ and $21$ .

$x^{4} - 17 x^{2} + 70 + 2 = 0$

Step 1.2.2.2

Add $70$ and $2$ .

$x^{4} - 17 x^{2} + 72 = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - 3, - 2 \sqrt{2}, 2 \sqrt{2}, 3$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - 3, - 2 \sqrt{2}, 2 \sqrt{2}, 3$

Decimal Form:

$x = - 3, - 2.82842712 \dots, 2.82842712 \dots, 3$

Step 4