Question

How do you factor the quadratic function $25{{x}^{2}}-10x+1$?

Answer 1

If $x=a$ is a root of a polynomial function, then $x-a$ is one of its factors. To express a quadratic equation $a{{x}^{2}}+bx+c$ in its factored form. We have to find its roots, say $\alpha ,\beta$ are the two real roots of the equation. Then the factored form is $a\left( x-\alpha \right)\left( x-\beta \right)$. We can find the roots of the equation using the quadratic formula method as $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.

Complete step by step solution:
We are given the quadratic expression $25{{x}^{2}}-10x+1$. On comparing with the general solution of the quadratic equation $a{{x}^{2}}+bx+c$, we get $a=25,b=-10\And c=1$.
To express in factored form, we first have to find the roots of the equation $25{{x}^{2}}-10x+1$.
We can find the roots of the equation using the formula method.
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substituting the values of the coefficients in the above formula, we get

& \Rightarrow x=\dfrac{-(-10)\pm \sqrt{{{\left( -10 \right)}^{2}}-4(25)(1)}}{2(25)} \\\ & \Rightarrow x=\dfrac{10\pm 0}{50} \\\ & \Rightarrow x=\dfrac{1}{5} \\\ \end{aligned}$$ $$\Rightarrow x=\alpha =\beta =\dfrac{1}{5}$$ Now that, we have the roots of the given expression, we can express it as its factored form as follows, For the quadratic expression $$25{{x}^{2}}-10x+1$$, $$a=25$$ and the roots as $$\alpha =\beta =\dfrac{1}{5}$$. The factored form is, $$\Rightarrow 25\left( x-\dfrac{1}{5} \right)\left( x-\dfrac{1}{5} \right)$$ Simplifying the above expression, and cancelling out common factors the expression can be expressed as $$\Rightarrow 25\left( x-\dfrac{1}{5} \right)\left( x-\dfrac{1}{5} \right)$$ $$\begin{aligned} & \Rightarrow 25{{\left( x-\dfrac{1}{5} \right)}^{2}} \\\ & \Rightarrow 25{{\left( \dfrac{5x-1}{5} \right)}^{2}} \\\ & \Rightarrow {{\left( 5x-1 \right)}^{2}} \\\ \end{aligned}$$ **Note:** The given equation has both roots same, thus we get a quadratic expression as its factored form. If the roots are different, we will get two separate linear expressions as the factored form. It should be noted that an expression can only be expressed as its factored form if it has real roots. For example, the quadratic expression $${{x}^{2}}+1$$ has no real roots. Hence, it can not be expressed as its factored form.