Solve for x 2x-7x^(1/2)=15

$2 x - 7 x^{\frac{1}{2}} = 15$

Find a common factor $x^{\frac{1}{2}}$ that is present in each term.

$2 {(x^{\frac{1}{2}})}^{2} - 7 x^{\frac{1}{2}} - 15$

Substitute $u$ for $x^{\frac{1}{2}}$ .

$2 {(u)}^{2} - 7 u - 15 = 0$

Solve for $u$ .

Remove parentheses.

$2 u^{2} - 7 u - 15 = 0$

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 $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 u$ .

$2 u^{2} - 7 u - 15 = 0$

Rewrite $- 7$ as $3$ plus $- 10$

$2 u^{2} + (3 - 10) u - 15 = 0$

Apply the distributive property.

$2 u^{2} + 3 u - 10 u - 15 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 u^{2} + 3 u) - 10 u - 15 = 0$

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

$u (2 u + 3) - 5 (2 u + 3) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 u + 3$ .

$(2 u + 3) (u - 5) = 0$

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

$2 u + 3 = 0$

$u - 5 = 0$

Set $2 u + 3$ equal to $0$ and solve for $u$ .

Set $2 u + 3$ equal to $0$ .

$2 u + 3 = 0$

Solve $2 u + 3 = 0$ for $u$ .

Subtract $3$ from both sides of the equation.

$2 u = - 3$

Divide each term in $2 u = - 3$ by $2$ and simplify.

Divide each term in $2 u = - 3$ by $2$ .

$\frac{2 u}{2} = \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 3}{2}$

Divide $u$ by $1$ .

$u = \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$u = - \frac{3}{2}$

Set $u - 5$ equal to $0$ and solve for $u$ .

Set $u - 5$ equal to $0$ .

$u - 5 = 0$

Add $5$ to both sides of the equation.

$u = 5$

The final solution is all the values that make $(2 u + 3) (u - 5) = 0$ true.

$u = - \frac{3}{2}, 5$

Substitute $x$ for $u$ .

$x^{\frac{1}{2}} = - \frac{3}{2}, 5$

Solve for $x^{\frac{1}{2}} = - \frac{3}{2}$ for $x$ .

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x = {(- \frac{3}{2})}^{2}$

Simplify the right side.

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{3}{2}$ .

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

Apply the product rule to $\frac{3}{2}$ .

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

Raise $- 1$ to the power of $2$ .

$x = 1 \frac{3^{2}}{2^{2}}$

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$x = \frac{3^{2}}{2^{2}}$

Raise $3$ to the power of $2$ .

$x = \frac{9}{2^{2}}$

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

$x = \frac{9}{4}$

Solve for $x^{\frac{1}{2}} = 5$ for $x$ .

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 5^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 5^{2}$

Rewrite the expression.

$x^{1} = 5^{2}$

Simplify.

$x = 5^{2}$

Simplify the right side.

Raise $5$ to the power of $2$ .

$x = 25$

List all of the solutions.

$x = \frac{9}{4}, 25$

Exclude the solutions that do not make $2 x - 7 x^{\frac{1}{2}} = 15$ true.

$x = 25$

Solve for x 2x-7x^(1/2)=15

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