Finite Math Examples

$u = \sqrt{2 u^{2} - 3 u - 4}$

Step 1

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

$\sqrt{2 u^{2} - 3 u - 4} = u$

Step 2

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

${\sqrt{2 u^{2} - 3 u - 4}}^{2} = u^{2}$

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 $\sqrt{2 u^{2} - 3 u - 4}$ as ${(2 u^{2} - 3 u - 4)}^{\frac{1}{2}}$ .

${({(2 u^{2} - 3 u - 4)}^{\frac{1}{2}})}^{2} = u^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${({(2 u^{2} - 3 u - 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Multiply the exponents in ${({(2 u^{2} - 3 u - 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1.1

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

${(2 u^{2} - 3 u - 4)}^{\frac{1}{2} \cdot 2} = u^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

${(2 u^{2} - 3 u - 4)}^{\frac{1}{2} \cdot 2} = u^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 u^{2} - 3 u - 4)}^{1} = u^{2}$

Step 3.2.1.2

Simplify.

$2 u^{2} - 3 u - 4 = u^{2}$

Step 4

Solve for $u$ .

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

$2 u^{2} - 3 u - 4 - u^{2} = 0$

Step 4.1.2

Subtract $u^{2}$ from $2 u^{2}$ .

$u^{2} - 3 u - 4 = 0$

Step 4.2

Factor $u^{2} - 3 u - 4$ using the AC method.

Tap for more steps...

Step 4.2.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 4.2.2

Write the factored form using these integers.

$(u - 4) (u + 1) = 0$

Step 4.3

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

$u - 4 = 0$

$u + 1 = 0$

Step 4.4

Set $u - 4$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 4.4.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 4.4.2

Add $4$ to both sides of the equation.

$u = 4$

Step 4.5

Set $u + 1$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 4.5.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 4.5.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 4.6

The final solution is all the values that make $(u - 4) (u + 1) = 0$ true.

$u = 4, - 1$

Step 5

Exclude the solutions that do not make $u = \sqrt{2 u^{2} - 3 u - 4}$ true.

$u = 4$