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Question: How do you solve \(2{{x}^{2}}-3x-6=0\) using the quadratic formula?...

How do you solve 2x2−3x−6=02{{x}^{2}}-3x-6=0 using the quadratic formula?

Explanation

Solution

We will look at the general quadratic equation. Then we will see the quadratic formula to solve this equation. We will compare the given equation with the general quadratic equation and obtain the corresponding coefficients. We will substitute these values in the quadratic formula and simplify the obtained expression. This will give us the required solution.

Complete step-by-step solution:
The general quadratic equation is given as ax2+bx+c=0a{{x}^{2}}+bx+c=0. The quadratic formula to obtain the solution of the general quadratic equation is given as the following,
x=−b±b2−4ac2ax=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}
The given quadratic equation is 2x2−3x−6=02{{x}^{2}}-3x-6=0. Comparing the given quadratic equation with the standard quadratic equation, we get the following as corresponding coefficients,
a=2a=2; b=−3b=-3 and c=−6c=-6.
Now, we will substitute the corresponding coefficients in the quadratic formula to obtain the solution for the given quadratic equation. Substituting the corresponding coefficients in the quadratic formula, we get the following expression,
x=−(−3)±(−3)2−4(2)(−6)2×2x=\dfrac{-\left( -3 \right)\pm \sqrt{{{\left( -3 \right)}^{2}}-4\left( 2 \right)\left( -6 \right)}}{2\times 2}
Simplifying the above expression, we get the following,
x=3±9+484 ∴x=3±574 \begin{aligned} & x=\dfrac{3\pm \sqrt{9+48}}{4} \\\ & \therefore x=\dfrac{3\pm \sqrt{57}}{4} \\\ \end{aligned}
Therefore, the solution of the given equation is x=3+574x=\dfrac{3+\sqrt{57}}{4} and x=3−574x=\dfrac{3-\sqrt{57}}{4}.

Note: There are other methods of solving a quadratic equation. These methods are the completing square method and the factorization method. We should choose the method according to our convenience if it is not mentioned in the question. This convenience greatly depends on the ease of calculation for these methods. We cannot take the square root of the number 57 since it is not a perfect square. The prime factors of 57 are 3 and 19. Therefore we cannot further simplify the square root term in the obtained solution.