Question

How do you factor \[4{{x}^{2}}-5x+7\]?

Answer 1

Hint: From the question given, we have been asked to factor the quadratic expression \[4{{x}^{2}}-5x+7\]. For that we will use the concept of determining the nature of the solutions of any quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ using the determinant given by $\Delta ={{b}^{2}}-4ac$ .

Complete step by step answer:
Now considering from the question we need to find the factors of $4{{x}^{2}}-5x+7$ .
From the basic concepts we know that we can determine the nature of the solutions of any quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ using the determinant given by $\Delta ={{b}^{2}}-4ac$ .
Here the value of the determinant will be
$\begin{align}
  & \Delta ={{\left( -5 \right)}^{2}}-4\left( 4 \right)\left( 7 \right) \\
 & \Rightarrow \Delta =25-112 \\
 & \Rightarrow \Delta =-87<0 \\
\end{align}$
Since $a=4$ , $b=-5$ and $c=7$ .
There are 4 cases for determine the nature of the solutions of any quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ using the determinant given by $\Delta ={{b}^{2}}-4ac$ . They are
Case 1: If $\Delta >0$ and a perfect square then there will be two rational solutions.
Case 2: If $\Delta >0$ and not a perfect square then there will be two irrational solutions.
Case 3: If $\Delta =0$ then there will be one solution.
Case 4: If $\Delta <0$ then there will be two complex conjugate solutions.
The solutions of the quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ using the formulae given as $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
Hence we will have only two complex conjugate solutions for the given expression.
Therefore, the given quadratic expression \[4{{x}^{2}}-5x+7\] cannot be factored using real numbers.
The solutions will be given as
$\begin{align}
  & \dfrac{-\left( -5 \right)\pm \sqrt{{{\left( -5 \right)}^{2}}-4\left( 4 \right)\left( 7 \right)}}{2\left( 4 \right)} \\
 & \Rightarrow \dfrac{5\pm \sqrt{-87}}{8} \\
 & \Rightarrow \dfrac{5\pm i\sqrt{87}}{8} \\
\end{align}$

Therefore the factors will be $x-\dfrac{5+i\sqrt{87}}{8}$ and $x-\dfrac{5-i\sqrt{87}}{8}$.

Note: We should be sure with our calculations and concept while answering questions of this type. We can also answer this question by following the traditional method of factorization but that will not be efficient. It will take away most of our team and still we will not be sure with our answer. The traditional method of factorization says to write the coefficient of $x$ as the sum of two numbers whose product will be equal to the constant term in the expression.