Solve by Factoring 2x-9 square root of x+4=0

$2 x - 9 \sqrt{x} + 4 = 0$

Step 1

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

$2 x - 9 x^{\frac{1}{2}} + 4 = 0$

Step 2

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

$2 {(x^{\frac{1}{2}})}^{2} - 9 x^{\frac{1}{2}} + 4 = 0$

Step 3

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

$2 u^{2} - 9 u + 4 = 0$

Step 4

Factor by grouping.

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 $a \cdot c = 2 \cdot 4 = 8$ and whose sum is $b = - 9$ .

Factor $- 9$ out of $- 9 u$ .

$2 u^{2} - 9 u + 4 = 0$

Rewrite $- 9$ as $- 1$ plus $- 8$

$2 u^{2} + (- 1 - 8) u + 4 = 0$

Apply the distributive property.

$2 u^{2} - 1 u - 8 u + 4 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 u^{2} - 1 u) - 8 u + 4 = 0$

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

$u (2 u - 1) - 4 (2 u - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$(2 u - 1) (u - 4) = 0$

Step 5

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

$(2 x^{\frac{1}{2}} - 1) (x^{\frac{1}{2}} - 4) = 0$

Step 6

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

$2 x^{\frac{1}{2}} - 1 = 0$

$x^{\frac{1}{2}} - 4 = 0$

Step 7

Set $2 x^{\frac{1}{2}} - 1$ equal to $0$ and solve for $x$ .

Set $2 x^{\frac{1}{2}} - 1$ equal to $0$ .

$2 x^{\frac{1}{2}} - 1 = 0$

Solve $2 x^{\frac{1}{2}} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x^{\frac{1}{2}} = 1$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(2 x^{\frac{1}{2}})}^{2} = 1^{2}$

Simplify the exponent.

Simplify the left side.

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

Apply the product rule to $2 x^{\frac{1}{2}}$ .

$2^{2} {(x^{\frac{1}{2}})}^{2} = 1^{2}$

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

$4 {(x^{\frac{1}{2}})}^{2} = 1^{2}$

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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

$4 x^{\frac{1}{2} \cdot 2} = 1^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x^{\frac{1}{2} \cdot 2} = 1^{2}$

Rewrite the expression.

$4 x^{1} = 1^{2}$

Simplify.

$4 x = 1^{2}$

Simplify the right side.

One to any power is one.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 8

Set $x^{\frac{1}{2}} - 4$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 4$ equal to $0$ .

$x^{\frac{1}{2}} - 4 = 0$

Solve $x^{\frac{1}{2}} - 4 = 0$ for $x$ .

Add $4$ to both sides of the equation.

$x^{\frac{1}{2}} = 4$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 4^{2}$

Simplify the exponent.

Simplify the left side.

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

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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

$x^{\frac{1}{2} \cdot 2} = 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 4^{2}$

Rewrite the expression.

$x^{1} = 4^{2}$

Simplify.

$x = 4^{2}$

Simplify the right side.

Raise $4$ to the power of $2$ .

$x = 16$

Step 9

The final solution is all the values that make $(2 x^{\frac{1}{2}} - 1) (x^{\frac{1}{2}} - 4) = 0$ true.

$x = \frac{1}{4}, 16$

Solve by Factoring 2x-9 square root of x+4=0

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