Question

How do you factor and solve $9{{x}^{2}}+7x-56=0$?

Answer 1

Hint: We have a quadratic equation which we will be solving using the discriminant. We see that, \[D>0\], so the expression has two real roots. Then, we will be using the quadratic formula to get the values of \[x\] by substituting the values of a, b and c in the formula. Hence, we will have two zeroes of the given expression.

Complete step by step solution:
According to the given question, we are given a quadratic equation, which we have to solve for \[x\]. If we try to factorize it directly, we see that it is not possible that way. So, we will be using the discriminant to solve the given expression and get the value of \[x\].
The given expression we have is,
\[9{{x}^{2}}+7x-56=0\]----(1)
And we get the values of \[a=9,b=7,c=-56\].
Discriminant, as we know,
\[D={{b}^{2}}-4ac\]
We will now substitute the known values in the formula of discriminant, we get,
\[\Rightarrow D={{(7)}^{2}}-4(9)(-56)\]
\[\Rightarrow D=49+2016\]
\[\Rightarrow D=2065>0\]
Since, we have the value of the discriminant greater than zero, the given expression has two real roots.
We will now use the quadratic formula to obtain the value of \[x\], we get,
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Substituting the values we know , we get,
\[\Rightarrow x=\dfrac{-7\pm \sqrt{2065}}{2(9)}\]
\[\Rightarrow x=\dfrac{-7\pm \sqrt{2065}}{18}\]
\[x=\dfrac{-7+\sqrt{2065}}{18},\dfrac{-7-\sqrt{2065}}{18}\]
Therefore, the value of \[x=\dfrac{-7+\sqrt{2065}}{18},\dfrac{-7-\sqrt{2065}}{18}\].

Note:
The discriminant calculated in the above solution should be done correctly and step wise only, so as to prevent any errors. The value of discriminant is substituted in the quadratic formula, so if the discriminant is calculated wrong, then the entire solution will get wrong. Also, while substituting the values of a, b and c in the quadratic formula, it should be done carefully without any mistakes.