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How do you solve \[{{x}^{2}}+5x-84=0\] using the quadratic formula?

Last updated date: 21st Jun 2024
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Hint: Compare the given quadratic equation with the general form given as: - \[a{{x}^{2}}+bx+c=0\]. Find the respective values of a, b and c. Now, find the discriminant of the given quadratic equation by using the formula: - \[D={{b}^{2}}-4ac\], where ‘D’ is the notation for the discriminant. Now, apply the formula: - \[x=\dfrac{-b\pm \sqrt{D}}{2a}\] and substitute the required values to get the answer.

Complete step by step solution:
Here, we have been provided with a quadratic equation: - \[{{x}^{2}}+5x-84=0\] and we are asked to solve it. That means we have to find the values of x. So, let us apply the discriminant method to solve the given quadratic equation.
Now, comparing the general form of a quadratic equation: - \[a{{x}^{2}}+bx+c=0\] with the given equation \[{{x}^{2}}+5x-84=0\], we can conclude that, we have,
\[\Rightarrow \] a = 1, b = 5 and c = -84.
Applying the formula for discriminant of a quadratic equation given as: - \[D={{b}^{2}}-4ac\], where ‘D’ is the discriminant, we get,
  & \Rightarrow D={{\left( 5 \right)}^{2}}-4\times 1\times \left( -84 \right) \\
 & \Rightarrow D=25+336 \\
 & \Rightarrow D=361 \\
Now, we know that the solution of a quadratic equation in terms of its discriminant value is given as: -
\[\Rightarrow x=\dfrac{-b\pm \sqrt{D}}{2a}\]
So, substituting the given values and obtained values of D, we get,
  & \Rightarrow x=\dfrac{-5\pm \sqrt{361}}{2\times 1} \\
 & \Rightarrow x=\dfrac{-5\pm 19}{2} \\
(i) Considering (+) sign we have,
  & \Rightarrow x=\dfrac{-5+19}{2} \\
 & \Rightarrow x=\dfrac{14}{2} \\
 & \Rightarrow x=7 \\
(ii) Considering (-) sign we have,
  & \Rightarrow x=\dfrac{-5-19}{2} \\
 & \Rightarrow x=\dfrac{-24}{2} \\
 & \Rightarrow x=-12 \\
Hence, the above two values of x are the roots or solution of the given quadratic equation.

One may note that we can also use the middle term split method to solve the question and check if we are getting the same answer or not. In that case we need to break the middle term 5x into two terms such that their sum equals 5x and product equals $-84{{x}^{2}}$. You may apply the third method known as completing the square method to solve the question. In this method we have to convert the equation \[a{{x}^{2}}+bx+c=0\] into the form: - \[{{\left( x+\dfrac{b}{2a} \right)}^{2}}=\dfrac{D}{4{{a}^{2}}}\] and then by taking the square root we need to solve for the value of x. Remember that the discriminant formula is derived from completing the square method. You must remember the discriminant formula to solve the above question.