Question

How do you solve \[{{x}^{2}}-10x=21\]?

Answer 1

Hint: From the question, we have been given that to solve \[{{x}^{2}}-10x=21\]. We can solve the given equation by transforming the given equation into a quadratic equation. Then, by the method of factorization or by using the quadratic formula, we can solve the given equation.

Complete step-by-step solution:
From the question, we have been given that \[{{x}^{2}}-10x=21\]
Now to transform the given equation into a quadratic equation shift \[21\] from the right hand side of the equation to the left hand side of the equation.
By shifting \[21\] from right hand side of the equation to the left hand side of the equation, we get \[{{x}^{2}}-10x-21=0\]
Clearly, we can observe that the equation is in the form of \[a{{x}^{2}}+bx+c=0\].
For this quadratic equation, the factorization process will not work. So by using the quadratic formula, we have to solve the obtained quadratic equation.
By comparing the coefficients of the obtained quadratic equation and general form of the quadratic equation, we get
\[\begin{align}
  & a=1 \\
 & b=-10 \\
 & c=-21 \\
\end{align}\]
Now, quadratic formula for the quadratic equation \[a{{x}^{2}}+bx+c=0\] is \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Now, substitute the values we got in the quadratic formula.
By substituting the values we got in the quadratic formula, we get
\[x=\dfrac{-\left( -10 \right)\pm \sqrt{{{\left( -10 \right)}^{2}}-4\left( 1 \right)\left( -21 \right)}}{2\left( 1 \right)}\]
\[\Rightarrow x=\dfrac{10\pm \sqrt{100+84}}{2}\]
\[\Rightarrow x=\dfrac{10\pm \sqrt{184}}{2}\]
\[\Rightarrow x=\dfrac{10\pm \sqrt{4\left( 46 \right)}}{2}\]
\[\Rightarrow x=\dfrac{10\pm 2\sqrt{\left( 46 \right)}}{2}\]
\[\Rightarrow x=\dfrac{2\left( 5\pm \sqrt{\left( 46 \right)} \right)}{2}\]
\[\Rightarrow x=5\pm \sqrt{46}\]
Therefore, \[x=5\pm \sqrt{46}\] are the solutions for the given question.
Hence, the given question is solved.

Note: We should be very careful while converting the given equation into the general form of quadratic equation. We should be very careful while doing the calculation of quadratic formulas. Also, we should check whether factorization is possible for the equation or not, if possible it is better to do factorization as it is an easy process. Similarly we can answer \[{{x}^{2}}-10x+21=0\Rightarrow \left( x-3 \right)\left( x-7 \right)=0\Rightarrow x=3,7\] .