Question

Solve :- $x^{\frac{1}{2}}+8x^{\frac{1}{4}}+15=0$

Short answer

Solution for equation will be -

• $x=625$
  
• $x=81$
  

Step-by-step solution

Step 1.  Given Data

Quadratic Equation - $x^{\frac{1}{2}}+8x^{\frac{1}{4}}+15=0$

Step 2.  To Find

We have to solve it and find the value of x.

Step 3.  Explanation

Let the value of $x^{\frac{1}{4}}=u$

Then the equation is -

$x^{\frac{1}{2}}+8x^{\frac{1}{4}}+15=0$

${\left(x^{\frac{1}{4}}\right)}^2+8x^{\frac{1}{4}}+15=0$

Put the value of $x^{\frac{1}{4}}=u$

$u^2+8u+15=0$

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

By using zero product property -

$u+5=0$, $u+3=0$

role="math" localid="1645027263305" $u=(-5)$, role="math" localid="1645027241340" $u=(-3)$

Step 4.  Calculation

Replace u with $x^{\frac{1}{4}}$

$x^{\frac{1}{4}}=(-5)$, $x^{\frac{1}{4}}=(-3)$

Solve for x by do power fourth both sides

${\left(x^{\frac{1}{4}}\right)}^4={(-5)}^4$, ${\left(x^{\frac{1}{4}}\right)}^4={(-3)}^4$

$x=624$, $x=81$