Finite Math Examples

$a d^{2} - 126 d + 49 = {(9 d - 7)}^{2}$

Step 1

Simplify ${(9 d - 7)}^{2}$ .

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Step 1.1

Rewrite.

$a d^{2} - 126 d + 49 = 0 + 0 + {(9 d - 7)}^{2}$

Step 1.2

Rewrite ${(9 d - 7)}^{2}$ as $(9 d - 7) (9 d - 7)$ .

$a d^{2} - 126 d + 49 = (9 d - 7) (9 d - 7)$

Step 1.3

Expand $(9 d - 7) (9 d - 7)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$a d^{2} - 126 d + 49 = 9 d (9 d - 7) - 7 (9 d - 7)$

Step 1.3.2

Apply the distributive property.

$a d^{2} - 126 d + 49 = 9 d (9 d) + 9 d \cdot - 7 - 7 (9 d - 7)$

Step 1.3.3

Apply the distributive property.

$a d^{2} - 126 d + 49 = 9 d (9 d) + 9 d \cdot - 7 - 7 (9 d) - 7 \cdot - 7$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$a d^{2} - 126 d + 49 = 9 \cdot 9 d \cdot d + 9 d \cdot - 7 - 7 (9 d) - 7 \cdot - 7$

Step 1.4.1.2

Multiply $d$ by $d$ by adding the exponents.

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Step 1.4.1.2.1

Move $d$ .

$a d^{2} - 126 d + 49 = 9 \cdot 9 (d \cdot d) + 9 d \cdot - 7 - 7 (9 d) - 7 \cdot - 7$

Step 1.4.1.2.2

Multiply $d$ by $d$ .

$a d^{2} - 126 d + 49 = 9 \cdot 9 d^{2} + 9 d \cdot - 7 - 7 (9 d) - 7 \cdot - 7$

Step 1.4.1.3

Multiply $9$ by $9$ .

$a d^{2} - 126 d + 49 = 81 d^{2} + 9 d \cdot - 7 - 7 (9 d) - 7 \cdot - 7$

Step 1.4.1.4

Multiply $- 7$ by $9$ .

$a d^{2} - 126 d + 49 = 81 d^{2} - 63 d - 7 (9 d) - 7 \cdot - 7$

Step 1.4.1.5

Multiply $9$ by $- 7$ .

$a d^{2} - 126 d + 49 = 81 d^{2} - 63 d - 63 d - 7 \cdot - 7$

Step 1.4.1.6

Multiply $- 7$ by $- 7$ .

$a d^{2} - 126 d + 49 = 81 d^{2} - 63 d - 63 d + 49$

Step 1.4.2

Subtract $63 d$ from $- 63 d$ .

$a d^{2} - 126 d + 49 = 81 d^{2} - 126 d + 49$

Step 2

Move all terms containing $d$ to the left side of the equation.

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Step 2.1

Subtract $81 d^{2}$ from both sides of the equation.

$a d^{2} - 126 d + 49 - 81 d^{2} = - 126 d + 49$

Step 2.2

Add $126 d$ to both sides of the equation.

$a d^{2} - 126 d + 49 - 81 d^{2} + 126 d = 49$

Step 2.3

Combine the opposite terms in $a d^{2} - 126 d + 49 - 81 d^{2} + 126 d$ .

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Step 2.3.1

Add $- 126 d$ and $126 d$ .

$a d^{2} + 49 - 81 d^{2} + 0 = 49$

Step 2.3.2

Add $a d^{2} + 49 - 81 d^{2}$ and $0$ .

$a d^{2} + 49 - 81 d^{2} = 49$

Step 3

Move all terms not containing $d$ to the right side of the equation.

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Step 3.1

Subtract $49$ from both sides of the equation.

$a d^{2} - 81 d^{2} = 49 - 49$

Step 3.2

Subtract $49$ from $49$ .

$a d^{2} - 81 d^{2} = 0$

Step 4

Factor $d^{2}$ out of $a d^{2} - 81 d^{2}$ .

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Step 4.1

Factor $d^{2}$ out of $a d^{2}$ .

$d^{2} a - 81 d^{2} = 0$

Step 4.2

Factor $d^{2}$ out of $- 81 d^{2}$ .

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

Step 4.3

Factor $d^{2}$ out of $d^{2} a + d^{2} \cdot - 81$ .

$d^{2} (a - 81) = 0$

Step 5

Divide each term in $d^{2} (a - 81) = 0$ by $a - 81$ and simplify.

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Step 5.1

Divide each term in $d^{2} (a - 81) = 0$ by $a - 81$ .

$\frac{d^{2} (a - 81)}{a - 81} = \frac{0}{a - 81}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $a - 81$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{d^{2} (a - 81)}{a - 81} = \frac{0}{a - 81}$

Step 5.2.1.2

Divide $d^{2}$ by $1$ .

$d^{2} = \frac{0}{a - 81}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $a - 81$ .

$d^{2} = 0$

Step 6

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$d = \pm \sqrt{0}$

Step 7

Simplify $\pm \sqrt{0}$ .

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Step 7.1

Rewrite $0$ as $0^{2}$ .

$d = \pm \sqrt{0^{2}}$

Step 7.2

Pull terms out from under the radical, assuming positive real numbers.

$d = \pm 0$

Step 7.3

Plus or minus $0$ is $0$ .

$d = 0$