Algebra Examples

2 x^{2} + 3 x - 20 = 0

$2 x^{2} + 3 x - 20 = 0$

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 = 2 \cdot - 20 = - 40

$a \cdot c = 2 \cdot - 20 = - 40$ and whose sum is

b = 3

$b = 3$ .

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Step 1.1.1

Factor

3

$3$ out of

3 x

$3 x$ .

2 x^{2} + 3 (x) - 20 = 0

$2 x^{2} + 3 (x) - 20 = 0$

Step 1.1.2

Rewrite

3

$3$ as

- 5

$- 5$ plus

8

$8$

2 x^{2} + (- 5 + 8) x - 20 = 0

$2 x^{2} + (- 5 + 8) x - 20 = 0$

Step 1.1.3

Apply the distributive property.

2 x^{2} - 5 x + 8 x - 20 = 0

$2 x^{2} - 5 x + 8 x - 20 = 0$

2 x^{2} - 5 x + 8 x - 20 = 0

$2 x^{2} - 5 x + 8 x - 20 = 0$

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.

(2 x^{2} - 5 x) + 8 x - 20 = 0

$(2 x^{2} - 5 x) + 8 x - 20 = 0$

Step 1.2.2

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

x (2 x - 5) + 4 (2 x - 5) = 0

$x (2 x - 5) + 4 (2 x - 5) = 0$

x (2 x - 5) + 4 (2 x - 5) = 0

$x (2 x - 5) + 4 (2 x - 5) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor,

2 x - 5

$2 x - 5$ .

(2 x - 5) (x + 4) = 0

$(2 x - 5) (x + 4) = 0$

(2 x - 5) (x + 4) = 0

$(2 x - 5) (x + 4) = 0$

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 - 5 = 0

$2 x - 5 = 0$

x + 4 = 0

$x + 4 = 0$

Step 3

Set

2 x - 5

$2 x - 5$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.1

Set

2 x - 5

$2 x - 5$ equal to

0

$0$ .

2 x - 5 = 0

$2 x - 5 = 0$

Step 3.2

Solve

2 x - 5 = 0

$2 x - 5 = 0$ for

x

$x$ .

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Step 3.2.1

Add

5

$5$ to both sides of the equation.

2 x = 5

$2 x = 5$

Step 3.2.2

Divide each term in

2 x = 5

$2 x = 5$ by

2

$2$ and simplify.

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Step 3.2.2.1

Divide each term in

2 x = 5

$2 x = 5$ by

2

$2$ .

\frac{2 x}{2} = \frac{5}{2}

$\frac{2 x}{2} = \frac{5}{2}$

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.

$\frac{2 x}{2} = \frac{5}{2}$

Step 3.2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{5}{2}$

Step 4

Set

$x + 4$ equal to

$0$ and solve for

$x$ .

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Step 4.1

Set

$x + 4$ equal to

$0$ .

$x + 4 = 0$

Step 4.2

Subtract

$4$ from both sides of the equation.

$x = - 4$

Step 5

The final solution is all the values that make

$(2 x - 5) (x + 4) = 0$ true.

$x = \frac{5}{2}, - 4$