Question

How do you solve the equation $2{{a}^{2}}=-6+8a$?

Answer 1

Hint: First we will convert the given equation in general quadratic equation form. Then we will solve the obtained quadratic equation by splitting the middle term method. We will split the middle term of the equation $a{{x}^{2}}+bx+c=0$ such that the product of two numbers is equal to $a\times c$ and sum of two numbers is equal to b.

Complete step-by-step solution:
We have been given an equation $2{{a}^{2}}=-6+8a$.
We have to solve the given equation.
First let us rearrange the terms to convert the given equation into standard form. Then we will get
$\Rightarrow 2{{a}^{2}}-8a+6=0$
Now, divide the whole equation by 2 to make the coefficient of a equal to 1, then we will get
$\Rightarrow {{a}^{2}}-4a+3=0$
Now, we will use the split middle term method. We have to find two numbers such as the product of two numbers is equal to $a\times c=3\times 1=3$ and their sum is equal to $b=4$
So we will use two numbers as 3 and 1.
So splitting the middle term we will get
$\Rightarrow {{a}^{2}}-\left( 3a+a \right)+3=0$
Now, simplifying the above obtained equation we will get
$\Rightarrow {{a}^{2}}-3a-a+3=0$
Now, taking the common terms out we will get
$\Rightarrow a\left( a-3 \right)-1\left( a-3 \right)=0$
Now, again taking common factors out we will get
$\Rightarrow \left( a-3 \right)\left( a-1 \right)=0$
Now, equating each factor to zero we will get
$\Rightarrow a=3,a=1$
Hence by solving the given equation we get the value of a as 3 and 1.

Note: The value of variables obtained by solving the quadratic equation is also known as the roots of the equation. The number of roots of a quadratic equation depends on the degree of the equation. We can also solve the quadratic equation by using the square method, quadratic formula method and factorization method.