Basic Math Examples

$2 \sqrt{5 c + 1} - 7 = c - 2$

Step 1

Move all terms not containing $2 \sqrt{5 c + 1}$ to the right side of the equation.

Tap for more steps...

Step 1.1

Add $7$ to both sides of the equation.

$2 \sqrt{5 c + 1} = c - 2 + 7$

Step 1.2

Add $- 2$ and $7$ .

$2 \sqrt{5 c + 1} = c + 5$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{5 c + 1})}^{2} = {(c + 5)}^{2}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 c + 1}$ as ${(5 c + 1)}^{\frac{1}{2}}$ .

${(2 {(5 c + 1)}^{\frac{1}{2}})}^{2} = {(c + 5)}^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${(2 {(5 c + 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Apply the product rule to $2 {(5 c + 1)}^{\frac{1}{2}}$ .

$2^{2} {({(5 c + 1)}^{\frac{1}{2}})}^{2} = {(c + 5)}^{2}$

Step 3.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(5 c + 1)}^{\frac{1}{2}})}^{2} = {(c + 5)}^{2}$

Step 3.2.1.3

Multiply the exponents in ${({(5 c + 1)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 {(5 c + 1)}^{\frac{1}{2} \cdot 2} = {(c + 5)}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.3.2.1

Cancel the common factor.

$4 {(5 c + 1)}^{\frac{1}{2} \cdot 2} = {(c + 5)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$4 {(5 c + 1)}^{1} = {(c + 5)}^{2}$

Step 3.2.1.4

Simplify.

$4 (5 c + 1) = {(c + 5)}^{2}$

Step 3.2.1.5

Apply the distributive property.

$4 (5 c) + 4 \cdot 1 = {(c + 5)}^{2}$

Step 3.2.1.6

Multiply.

Tap for more steps...

Step 3.2.1.6.1

Multiply $5$ by $4$ .

$20 c + 4 \cdot 1 = {(c + 5)}^{2}$

Step 3.2.1.6.2

Multiply $4$ by $1$ .

$20 c + 4 = {(c + 5)}^{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify ${(c + 5)}^{2}$ .

Tap for more steps...

Step 3.3.1.1

Rewrite ${(c + 5)}^{2}$ as $(c + 5) (c + 5)$ .

$20 c + 4 = (c + 5) (c + 5)$

Step 3.3.1.2

Expand $(c + 5) (c + 5)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.2.1

Apply the distributive property.

$20 c + 4 = c (c + 5) + 5 (c + 5)$

Step 3.3.1.2.2

Apply the distributive property.

$20 c + 4 = c \cdot c + c \cdot 5 + 5 (c + 5)$

Step 3.3.1.2.3

Apply the distributive property.

$20 c + 4 = c \cdot c + c \cdot 5 + 5 c + 5 \cdot 5$

Step 3.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.3.1.1

Multiply $c$ by $c$ .

$20 c + 4 = c^{2} + c \cdot 5 + 5 c + 5 \cdot 5$

Step 3.3.1.3.1.2

Move $5$ to the left of $c$ .

$20 c + 4 = c^{2} + 5 \cdot c + 5 c + 5 \cdot 5$

Step 3.3.1.3.1.3

Multiply $5$ by $5$ .

$20 c + 4 = c^{2} + 5 c + 5 c + 25$

Step 3.3.1.3.2

Add $5 c$ and $5 c$ .

$20 c + 4 = c^{2} + 10 c + 25$

Step 4

Solve for $c$ .

Tap for more steps...

Step 4.1

Since $c$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$c^{2} + 10 c + 25 = 20 c + 4$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Subtract $20 c$ from both sides of the equation.

$c^{2} + 10 c + 25 - 20 c = 4$

Step 4.2.2

Subtract $20 c$ from $10 c$ .

$c^{2} - 10 c + 25 = 4$

Step 4.3

Subtract $4$ from both sides of the equation.

$c^{2} - 10 c + 25 - 4 = 0$

Step 4.4

Subtract $4$ from $25$ .

$c^{2} - 10 c + 21 = 0$

Step 4.5

Factor $c^{2} - 10 c + 21$ using the AC method.

Tap for more steps...

Step 4.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 4.5.2

Write the factored form using these integers.

$(c - 7) (c - 3) = 0$

Step 4.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$c - 7 = 0$

$c - 3 = 0$

Step 4.7

Set $c - 7$ equal to $0$ and solve for $c$ .

Tap for more steps...

Step 4.7.1

Set $c - 7$ equal to $0$ .

$c - 7 = 0$

Step 4.7.2

Add $7$ to both sides of the equation.

$c = 7$

Step 4.8

Set $c - 3$ equal to $0$ and solve for $c$ .

Tap for more steps...

Step 4.8.1

Set $c - 3$ equal to $0$ .

$c - 3 = 0$

Step 4.8.2

Add $3$ to both sides of the equation.

$c = 3$

Step 4.9

The final solution is all the values that make $(c - 7) (c - 3) = 0$ true.

$c = 7, 3$