Algebra Examples

$12 x - 5 \sqrt{x} - 2$

Step 1

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

$12 x - 5 x^{\frac{1}{2}} - 2$

Step 2

Rewrite $x$ as ${(x^{\frac{1}{2}})}^{2}$ .

$12 {(x^{\frac{1}{2}})}^{2} - 5 x^{\frac{1}{2}} - 2$

Step 3

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

$12 u^{2} - 5 u - 2$

Step 4

Factor by grouping.

Tap for more steps...

Step 4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot - 2 = - 24$ and whose sum is $b = - 5$ .

Tap for more steps...

Step 4.1.1

Factor $- 5$ out of $- 5 u$ .

$12 u^{2} - 5 (u) - 2$

Step 4.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$12 u^{2} + (3 - 8) u - 2$

Step 4.1.3

Apply the distributive property.

$12 u^{2} + 3 u - 8 u - 2$

Step 4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1

Group the first two terms and the last two terms.

$(12 u^{2} + 3 u) - 8 u - 2$

Step 4.2.2

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

$3 u (4 u + 1) - 2 (4 u + 1)$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $4 u + 1$ .

$(4 u + 1) (3 u - 2)$

Step 5

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$(4 x^{\frac{1}{2}} + 1) (3 x^{\frac{1}{2}} - 2)$