Question

Solve the given quadratic equation for x:
$8{{x}^{2}}-22x-21=0$

Answer 1

- Hint: We can solve the above quadratic equation by factorization method. In the factorization method, we will split the term 22x in such a way that the split terms of 22x on multiplication will yield $21\times 8$.

Complete step-by-step solution -

The equation given in the question that we have to solve is:
$8{{x}^{2}}-22x-21=0$
We are going to split 22 in 22x in such a way that the terms in the split on multiplication yield $21\times 8$.
Now, the factors of $21\times 8$ are $7\times 3\times 4\times 2$.
From the factors of $21\times 8$ we can write $21\times 8$ as $28\times 6$.
If we subtract 6 from 28 we will get 22 which is the coefficient of x in the given quadratic equation.
Splitting $22x$ as:
$22x=28x-6x$
Substituting the above value of $22x$ in the given quadratic equation we get,
$\begin{aligned} & 8{{x}^{2}}-22x-21=0 \\\ & \Rightarrow 8{{x}^{2}}-\left( 28x-6x \right)-21=0 \\\ & \Rightarrow 8{{x}^{2}}-28x+6x-21=0 \\\ & \Rightarrow 4x\left( 2x-7 \right)+3\left( 2x-7 \right)=0 \\\ & \Rightarrow \left( 4x+3 \right)\left( 2x-7 \right)=0 \\\ \end{aligned}$
Equating $4x+3$ equal to 0 and $2x-7$ equal to 0 we get,
$\begin{aligned} & 4x+3=0;2x-7=0 \\\ & \Rightarrow x=-\dfrac{3}{4},\dfrac{7}{2} \\\ \end{aligned}$
Hence, the solution of the given quadratic equation is $x=-\dfrac{3}{4},\dfrac{7}{2}$.

Note: The alternative way of solving the above quadratic equation as follows:
$8{{x}^{2}}-22x-21=0$
We are solving the roots of the equation by discriminant formula.
Discriminant of a quadratic equation is denoted by D.
For the quadratic equation $a{{x}^{2}}+bx+c=0$ the value of D is:
$D={{b}^{2}}-4ac$
Comparing this value of D with the given quadratic equation $8{{x}^{2}}-22x-21=0$ we get,
$\begin{aligned} & D={{\left( -22 \right)}^{2}}-4\left( 8 \right)\left( -21 \right) \\\ & \Rightarrow D=484+672 \\\ & \Rightarrow D=1156 \\\ \end{aligned}$
The discriminant formula for finding the roots of quadratic equation $a{{x}^{2}}+bx+c=0$ is:
$x=\dfrac{-b\pm \sqrt{D}}{2a}$
Comparing the above value of x with the given quadratic equation $8{{x}^{2}}-22x-21=0$ we get,
$\begin{aligned} & x=\dfrac{22\pm \sqrt{1156}}{16} \\\ & \Rightarrow x=\dfrac{22\pm 34}{16} \\\ \end{aligned}$
Taking plus sign we get the value of x as:
$\begin{aligned} & x=\dfrac{22+34}{16} \\\ & \Rightarrow x=\dfrac{56}{16}=\dfrac{7}{2} \\\ \end{aligned}$
Taking minus sign we get the value of x as:
$\begin{aligned} & x=\dfrac{22-34}{16} \\\ & \Rightarrow x=-\dfrac{12}{16}=-\dfrac{3}{4} \\\ \end{aligned}$
Hence, we have got the same values of x as that we have obtained in solution part of the question.