# How do You Solve for “x” in a Quadratic Equation?

## Algebra Math Problem: Find the value of *x* from the quadratic equation 8*x*^{3/2n} *−* 8*x*^{−3/2n} = 63

## Solution

Multiply by *x*^{3/2n} and transpose; thus

8 *x*^{3/2n} *−* 8 *x*^{−3/2n} = 8 *x*^{9/4n²} *−* 8* *=* *63 *x*^{3/2n}

⇒ 8 *x*^{9/4n²} *−* 63 *x*^{3/2n} *−* 8* *=* *0

Let *x*^{3/2n} = *t* , then

8 *x*^{9/4n²} *−* 63 *x*^{3/2n} *−* 8 = 8 *x*^{(3/2n)²} *−* 63 *x*^{3/2n} *−* 8 = 0

⇒ 8 *x*^{(3/2n)²} *−* 63 *x*^{3/2n} = 8*t*² *−* 63*t* *−* 8 = 0

8*t*² *−* 63*t* *−* 8 is a quartratic equation so

\begin{aligned} t&=\dfrac{63 \pm \sqrt{63^2-4\times 8 \times(-8)}}{2 \times 8} \\ \\ &=\dfrac{63 \pm \sqrt{3969+256}}{16} \\ \\ &=\dfrac{63 \pm \sqrt{4225}}{16} \\ \\ &=\dfrac{63 \pm 65}{16} \\ \\ \Rightarrow t&=8 , \ \dfrac{1}{8}\\ \end{aligned}

When *t* = 8

*x*^{3/2n} = 8

⇒ *x* = 8^{2n/3}

⇒ *x* = 4^{n}

When *t* = 1/8

*x*^{3/2n} = 1/8

⇒ *x* = 8^{−2n/3}

⇒ *x* = 4^{−n}

Thus, solution to this quadratic equation is *x =***4 ^{n}** and

*x*= 4^{−n}