Pre-Algebra Examples

8 x^{2} + 10 x - 7 = 0

$8 x^{2} + 10 x - 7 = 0$

Step 1

Factor by grouping.

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Step 1.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 8 \cdot - 7 = - 56

$a \cdot c = 8 \cdot - 7 = - 56$ and whose sum is

b = 10

$b = 10$ .

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Step 1.1.1

Factor

10

$10$ out of

10 x

$10 x$ .

8 x^{2} + 10 (x) - 7 = 0

$8 x^{2} + 10 (x) - 7 = 0$

Step 1.1.2

Rewrite

10

$10$ as

- 4

$- 4$ plus

14

$14$

8 x^{2} + (- 4 + 14) x - 7 = 0

$8 x^{2} + (- 4 + 14) x - 7 = 0$

Step 1.1.3

Apply the distributive property.

8 x^{2} - 4 x + 14 x - 7 = 0

$8 x^{2} - 4 x + 14 x - 7 = 0$

8 x^{2} - 4 x + 14 x - 7 = 0

$8 x^{2} - 4 x + 14 x - 7 = 0$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

(8 x^{2} - 4 x) + 14 x - 7 = 0

$(8 x^{2} - 4 x) + 14 x - 7 = 0$

Step 1.2.2

Factor out the greatest common factor (GCF) from each group.

4 x (2 x - 1) + 7 (2 x - 1) = 0

$4 x (2 x - 1) + 7 (2 x - 1) = 0$

4 x (2 x - 1) + 7 (2 x - 1) = 0

$4 x (2 x - 1) + 7 (2 x - 1) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor,

2 x - 1

$2 x - 1$ .

(2 x - 1) (4 x + 7) = 0

$(2 x - 1) (4 x + 7) = 0$

(2 x - 1) (4 x + 7) = 0

$(2 x - 1) (4 x + 7) = 0$

Step 2

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

2 x - 1 = 0

$2 x - 1 = 0$

4 x + 7 = 0

$4 x + 7 = 0$

Step 3

Set

2 x - 1

$2 x - 1$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.1

Set

2 x - 1

$2 x - 1$ equal to

0

$0$ .

2 x - 1 = 0

$2 x - 1 = 0$

Step 3.2

Solve

2 x - 1 = 0

$2 x - 1 = 0$ for

x

$x$ .

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Step 3.2.1

Add

1

$1$ to both sides of the equation.

2 x = 1

$2 x = 1$

Step 3.2.2

Divide each term in

2 x = 1

$2 x = 1$ by

2

$2$ and simplify.

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Step 3.2.2.1

Divide each term in

2 x = 1

$2 x = 1$ by

2

$2$ .

\frac{2 x}{2} = \frac{1}{2}

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of

2

$2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

\frac{2 x}{2} = \frac{1}{2}

$\frac{2 x}{2} = \frac{1}{2}$

Step 3.2.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{1}{2}

$x = \frac{1}{2}$

x = \frac{1}{2}

$x = \frac{1}{2}$

x = \frac{1}{2}

$x = \frac{1}{2}$

x = \frac{1}{2}

$x = \frac{1}{2}$

x = \frac{1}{2}

$x = \frac{1}{2}$

x = \frac{1}{2}

$x = \frac{1}{2}$

Step 4

Set

4 x + 7

$4 x + 7$ equal to

0

$0$ and solve for

x

$x$ .

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Step 4.1

Set

4 x + 7

$4 x + 7$ equal to

0

$0$ .

4 x + 7 = 0

$4 x + 7 = 0$

Step 4.2

Solve

4 x + 7 = 0

$4 x + 7 = 0$ for

x

$x$ .

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Step 4.2.1

Subtract

7

$7$ from both sides of the equation.

4 x = - 7

$4 x = - 7$

Step 4.2.2

Divide each term in

4 x = - 7

$4 x = - 7$ by

4

$4$ and simplify.

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Step 4.2.2.1

Divide each term in

4 x = - 7

$4 x = - 7$ by

4

$4$ .

\frac{4 x}{4} = \frac{- 7}{4}

$\frac{4 x}{4} = \frac{- 7}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of

4

$4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

\frac{4 x}{4} = \frac{- 7}{4}

$\frac{4 x}{4} = \frac{- 7}{4}$

Step 4.2.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 7}{4}

$x = \frac{- 7}{4}$

x = \frac{- 7}{4}

$x = \frac{- 7}{4}$

x = \frac{- 7}{4}

$x = \frac{- 7}{4}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

x = - \frac{7}{4}

$x = - \frac{7}{4}$

x = - \frac{7}{4}

$x = - \frac{7}{4}$

x = - \frac{7}{4}

$x = - \frac{7}{4}$

x = - \frac{7}{4}

$x = - \frac{7}{4}$

x = - \frac{7}{4}

$x = - \frac{7}{4}$

Step 5

The final solution is all the values that make

(2 x - 1) (4 x + 7) = 0

$(2 x - 1) (4 x + 7) = 0$ true.

x = \frac{1}{2}, - \frac{7}{4}

$x = \frac{1}{2}, - \frac{7}{4}$