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Question: How do you factor \(80{{x}^{2}}+68x+12\) ?...

How do you factor 80x2+68x+1280{{x}^{2}}+68x+12 ?

Explanation

Solution

In this question, we have to find the factors of the given equation. The equation given to us is in the form of a quadratic. Thus, we will apply the discriminant method to solve this problem. We compare the general form of quadratic equation and the given equation to get the value of a, b, and c. Then, we will find the value of discriminant using the formula D=b2−4acD=\sqrt{{{b}^{2}}-4ac}, and thus find the value of x using the discriminant formula x=−b±b2−4ac2ax=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} .After the necessary calculations, we get two equations for x, thus we solve them separately, which is our required answer.

Complete step by step solution:
According to the question, we have to find the factor of a quadratic equation.
Thus, we will use the discriminant method to get the solution.
The equation is 80x2+68x+1280{{x}^{2}}+68x+12 ----------------- (1)
As we know, the general quadratic equation is in form of ay2+by+c=0a{{y}^{2}}+by+c=0 ---------- (2)
Thus, on comparing equation (1) and (2), we get a=80,a=80, b=68,b=68, and c=12c=12 ------- (3)
So, now we will find the value of discriminant using the formula D=b2−4acD=\sqrt{{{b}^{2}}-4ac} by putting the above values in the formula, we get
⇒D=(68)2−4.(80).(12) ⇒D=4624−3840 \begin{aligned} & \Rightarrow D=\sqrt{{{(68)}^{2}}-4.(80).(12)} \\\ & \Rightarrow D=\sqrt{4624-3840} \\\ \end{aligned}
Thus, on further solving, we get
⇒D=784\Rightarrow D=\sqrt{784}
Since D>0 which implies the equation has real roots.
⇒D=28\Rightarrow D=28 -------------- (4)
Since we see the discriminant is a real number, thus now we will find the value of x, using the formula,
⇒x=−b±D2a\Rightarrow x=\dfrac{-b\pm D}{2a}
⇒x=−b±b2−4ac2a\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} --------------- (5)
So, we will put the value of equation (3) and (4) in equation (5), we get
⇒x=−(68)±282.(80)\Rightarrow x=\dfrac{-\left( 68 \right)\pm 28}{2.\left( 80 \right)}
On further simplification, we get
⇒x=−68±28160\Rightarrow x=\dfrac{-68\pm 28}{160}
Therefore, we will split the above equation in terms of (+) and (-), we get
⇒x=−68+28160\Rightarrow x=\dfrac{-68+28}{160} -------- (6) , and
⇒x=−68−28160\Rightarrow x=\dfrac{-68-28}{160} ---------- (7)
Now, we will first solve equation (6), we get
⇒x=−40160\Rightarrow x=\dfrac{-40}{160}
Therefore, we get
⇒x=−14\Rightarrow x=\dfrac{-1}{4}
Now we will solve equation (7), we get
⇒x=−96160\Rightarrow x=\dfrac{-96}{160}
Therefore, we get
⇒x=−35\Rightarrow x=\dfrac{-3}{5}
Therefore, for the quadratic equation 80x2+68x+1280{{x}^{2}}+68x+12 , its factors is equal to (x+14)(x+35)\left( x+\dfrac{1}{4} \right)\left( x+\dfrac{3}{5} \right)

Note: While solving this problem, do step-by-step calculations carefully to avoid mathematical errors to get the correct answer. One of the alternative methods for solving this problem is using the split middle term method or the cross multiplication method to get the solution.