Basic Math Examples

4 a^{2} + 15 = 16 a

$4 a^{2} + 15 = 16 a$

Step 1

Subtract

16 a

$16 a$ from both sides of the equation.

4 a^{2} + 15 - 16 a = 0

$4 a^{2} + 15 - 16 a = 0$

Step 2

Factor by grouping.

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Step 2.1

Reorder terms.

4 a^{2} - 16 a + 15 = 0

$4 a^{2} - 16 a + 15 = 0$

Step 2.2

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 = 4 \cdot 15 = 60

$a \cdot c = 4 \cdot 15 = 60$ and whose sum is

b = - 16

$b = - 16$ .

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Step 2.2.1

Factor

- 16

$- 16$ out of

- 16 a

$- 16 a$ .

4 a^{2} - 16 a + 15 = 0

$4 a^{2} - 16 a + 15 = 0$

Step 2.2.2

Rewrite

- 16

$- 16$ as

- 6

$- 6$ plus

- 10

$- 10$

4 a^{2} + (- 6 - 10) a + 15 = 0

$4 a^{2} + (- 6 - 10) a + 15 = 0$

Step 2.2.3

Apply the distributive property.

4 a^{2} - 6 a - 10 a + 15 = 0

$4 a^{2} - 6 a - 10 a + 15 = 0$

4 a^{2} - 6 a - 10 a + 15 = 0

$4 a^{2} - 6 a - 10 a + 15 = 0$

Step 2.3

Factor out the greatest common factor from each group.

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Step 2.3.1

Group the first two terms and the last two terms.

(4 a^{2} - 6 a) - 10 a + 15 = 0

$(4 a^{2} - 6 a) - 10 a + 15 = 0$

Step 2.3.2

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

2 a (2 a - 3) - 5 (2 a - 3) = 0

$2 a (2 a - 3) - 5 (2 a - 3) = 0$

2 a (2 a - 3) - 5 (2 a - 3) = 0

$2 a (2 a - 3) - 5 (2 a - 3) = 0$

Step 2.4

Factor the polynomial by factoring out the greatest common factor,

2 a - 3

$2 a - 3$ .

(2 a - 3) (2 a - 5) = 0

$(2 a - 3) (2 a - 5) = 0$

(2 a - 3) (2 a - 5) = 0

$(2 a - 3) (2 a - 5) = 0$

Step 3

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 a - 3 = 0

$2 a - 3 = 0$

2 a - 5 = 0

$2 a - 5 = 0$

Step 4

Set

2 a - 3

$2 a - 3$ equal to

0

$0$ and solve for

a

$a$ .

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Step 4.1

Set

2 a - 3

$2 a - 3$ equal to

0

$0$ .

2 a - 3 = 0

$2 a - 3 = 0$

Step 4.2

Solve

2 a - 3 = 0

$2 a - 3 = 0$ for

a

$a$ .

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Step 4.2.1

Add

3

$3$ to both sides of the equation.

2 a = 3

$2 a = 3$

Step 4.2.2

Divide each term in

2 a = 3

$2 a = 3$ by

2

$2$ and simplify.

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Step 4.2.2.1

Divide each term in

2 a = 3

$2 a = 3$ by

2

$2$ .

\frac{2 a}{2} = \frac{3}{2}

$\frac{2 a}{2} = \frac{3}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of

2

$2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{3}{2}$

Step 4.2.2.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{3}{2}$

Step 5

Set

$2 a - 5$ equal to

$0$ and solve for

$a$ .

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Step 5.1

Set

$2 a - 5$ equal to

$0$ .

$2 a - 5 = 0$

Step 5.2

Solve

$2 a - 5 = 0$ for

$a$ .

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Step 5.2.1

Add

$5$ to both sides of the equation.

$2 a = 5$

Step 5.2.2

Divide each term in

$2 a = 5$ by

$2$ and simplify.

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Step 5.2.2.1

Divide each term in

$2 a = 5$ by

$2$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of

$2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide

$a$ by

$1$ .

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

Step 6

The final solution is all the values that make

$(2 a - 3) (2 a - 5) = 0$ true.

$a = \frac{3}{2}, \frac{5}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{3}{2}, \frac{5}{2}$

Decimal Form:

$a = 1.5, 2.5$

Mixed Number Form:

$a = 1 \frac{1}{2}, 2 \frac{1}{2}$