Question

Solve it. $8{{x}^{2}}-10x-3=0.$

Answer 1

Hint: We solve this question by splitting the middle term. The given equation is a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ and we need to split the middle term into two terms such that its sum is equal to $b$ and its product is equal to $a\times c.$

Complete step by step solution:
In order to solve this question, let us consider the equation given in the question $8{{x}^{2}}-10x-3=0.$ This is of the form of a general quadratic equation given by $a{{x}^{2}}+bx+c=0.$ Here, a, b and c are coefficients of the quadratic equation. We need to split the middle coefficient term in the given equation which is $-10$ into two numbers.
These numbers should be such that their product is equal to $a\times c$ and their sum is equal to $b.$ According to the given equation, the product of the terms is $a\times c=8\times -3=-24.$ The sum of the two numbers is $b=-10.$ The two numbers can be split as $-12$ and $+2$ since their sum is $-10$ and their product is $-24.$
Consider the given equation,
$\Rightarrow 8{{x}^{2}}-10x-3=0$
We rewrite the equation by splitting the middle term into the two terms $-12$ and $+2$ as follows,
$\Rightarrow 8{{x}^{2}}-12x+2x-3=0$
Now, we take $4x$ common out from the first two terms and since there is nothing to take out from the last two terms, we just group them together by taking 1 common out.
$\Rightarrow 4x\left( 2x-3 \right)+1\left( 2x-3 \right)=0$
Now we take the term $\left( 2x-3 \right)$ common out from the given two terms.
$\Rightarrow \left( 2x-3 \right)\left( 4x+1 \right)=0$
Since the product of these two terms is equal to zero, we individually equate them to 0. Taking the first term equal to 0.
$\Rightarrow \left( 2x-3 \right)=0$
Adding 3 on both sides,
$\Rightarrow 2x=3$
Dividing both sides of the equation by 2,
$\Rightarrow x=\dfrac{3}{2}$
Similarly, we equate the second term to 0.
$\Rightarrow \left( 4x+1 \right)=0$
Subtracting both sides by 1,
$\Rightarrow 4x=-1$
Dividing both sides by 4,
$\Rightarrow x=-\dfrac{1}{4}$

Hence, by solving the given equation $8{{x}^{2}}-10x-3=0,$ we get the solution by obtaining the values for x as $x=\dfrac{3}{2}$ and $x=-\dfrac{1}{4}.$

Note:
We need to know the basics of quadratic equations to solve this sum. We can also solve this question by making the coefficient of ${{x}^{2}}$ term 1 by dividing the entire equation by 8 and solving by splitting the middle term in the same way as given above. We can also use other methods like completing the square method in order to solve this equation.