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Question: How do you solve \(-3{{x}^{2}}=x-6\) using the quadratic formula?...

How do you solve −3x2=x−6-3{{x}^{2}}=x-6 using the quadratic formula?

Explanation

Solution

For this question, a quadratic equation is an equation which is in the form of ax2+bx+c=0a{{x}^{2}}+bx+c=0 and has two roots. We need to find the roots or zeros for this equation. For this, you should know that roots are those values which satisfy the equation and result in zero when they are placed and solved in the equation. To find roots, quadratic formula is used:
⇒x=−b±b2−4ac2a\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}

Complete step-by-step solution:
Now, let’s solve the question.
As we are already aware that the quadratic equation is an equation which has the 2 as its highest degree and is in the form of ax2+bx+c=0a{{x}^{2}}+bx+c=0. Here ‘a’ and ‘b’ are coefficients of x2{{x}^{2}} and x respectively, and c is the constant term. It consists of two roots which satisfy the equation and results in zero after placing the roots in an equation. And you know that there are basically two methods of finding roots of an equation. The two methods are middle term splitting and by using quadratic formula. So, we will be using quadratic formula for this equation because middle term splitting is only possible if we can split the middle term into two terms and we are getting something common while grouping them in pairs as we are not be able to split its middle term. Quadratic formula is:
⇒x=−b±b2−4ac2a\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}
Now, write the given equation.
⇒−3x2=x−6\Rightarrow -3{{x}^{2}}=x-6
Convert the equation in the form of ax2+bx+c=0a{{x}^{2}}+bx+c=0:
⇒−3x2−x+6=0\Rightarrow -3{{x}^{2}}-x+6=0
Take negative common:
⇒3x2+x−6=0\Rightarrow 3{{x}^{2}}+x-6=0
From the equation above, a = 3, b = 1 and c = -6. Place all the values in quadratic formula, we will get:
⇒x=−1±(1)2−4(3)(−6)2(1)\Rightarrow x=\dfrac{-1\pm \sqrt{{{\left( 1 \right)}^{2}}-4\left( 3 \right)\left( -6 \right)}}{2\left( 1 \right)}
On solving further:
⇒x=−1±1+722\Rightarrow x=\dfrac{-1\pm \sqrt{1+72}}{2}
On Solving the under root:
⇒x=−1±732\Rightarrow x=\dfrac{-1\pm \sqrt{73}}{2}
Roots for x are:
**⇒x=−1+732,−1−732\Rightarrow x=\dfrac{-1+\sqrt{73}}{2},\dfrac{-1-\sqrt{73}}{2}
This is the final answer. **

Note: Here, root of 73 cannot be found as 73 is a prime number so we cannot split and find it’s under root. That’s why it is taken as 73\sqrt{73}itself. Although it is mentioned to solve the equation by quadratic formula but otherwise you can first try to solve by the middle term splitting because it is the easiest approach for any quadratic equation.