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Question: How do you factor \({{x}^{2}}+5x-36\) ?...

How do you factor x2+5x−36{{x}^{2}}+5x-36 ?

Explanation

Solution

From the question given, we have been asked to factor the quadratic expression x2+5x−36{{x}^{2}}+5x-36. We can find the factors for the given quadratic expression by using the process of factorization. By using the process of factorization, we can find the factors for the given quadratic expression. First of all, we have to know about the process of factorization.

Complete step-by-step solution:
By using the product of coefficients and writing it as the sum of the two numbers we will find out the factors. This is the process of factorization.
From the question given, we have been given a quadratic expression x2+5x−36{{x}^{2}}+5x-36
First of all, we have to multiply the coefficient of x2{{x}^{2}} and constant.
By multiplying the coefficient of x2{{x}^{2}} and constant, we get ⇒1×−36=−36\Rightarrow 1\times -36=-36
Now, we have to write the coefficient of xx as the sum of two numbers and the numbers should be the factors of 3636.
⇒−36=9×−4\Rightarrow -36=9\times -4
By writing the given expression as above said, we get ⇒x2+9x−4x−36\Rightarrow {{x}^{2}}+9x-4x-36
Now, we have to take the common terms out and simplify further to obtain the factors.
By taking the common terms out and simplifying further, we get
⇒x(x+9)−4(x+9)\Rightarrow x\left( x+9 \right)-4\left( x+9 \right)
⇒(x−4)(x+9)\Rightarrow \left( x-4 \right)\left( x+9 \right)
Therefore, x2+5x−36=(x−4)(x+9){{x}^{2}}+5x-36=\left( x-4 \right)\left( x+9 \right)
Hence, we got the factors for the given quadratic expression by using the process of factorization.

Note: We should be well aware of the process of factorization. Also, we should be very careful while writing the middle term as a sum of two numbers which is equal to the product of coefficient of first term and constant. Also, we should be very careful while simplifying the expression. This question can also be answered by using the formulae for finding the roots of any quadratic equation in the form of ax2+bx+c=0a{{x}^{2}}+bx+c=0 is given by −b±b2−4ac2a\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} . For x2+5x−36{{x}^{2}}+5x-36 it is given as
−(5)±(5)2−4(1)(−36)2(1)=5±52+1442 ⇒−5±25+1442=−5±1692 ⇒−5±132=−9,4 \begin{aligned} & \dfrac{-\left( 5 \right)\pm \sqrt{{{\left( 5 \right)}^{2}}-4\left( 1 \right)\left( -36 \right)}}{2\left( 1 \right)}=\dfrac{5\pm \sqrt{{{5}^{2}}+144}}{2} \\\ & \Rightarrow \dfrac{-5\pm \sqrt{25+144}}{2}=\dfrac{-5\pm \sqrt{169}}{2} \\\ & \Rightarrow \dfrac{-5\pm 13}{2}= -9,4 \\\ \end{aligned}
As the roots are 9,−49,-4 the factors will be (x−4)\left( x-4 \right) and (x+9)\left( x+9 \right).