Question

How do you factor and solve ${x^2} + 100 = 20x?$

Answer 1

Hint: In this question we are going to factor and solve the given expression and then find the values of $x$
First write the given equation in a quadratic form.
In this we are going to factor the expression by splitting the middle term.
Multiply the coefficient of the first term by a constant in the given expression, we get a number and then find two factors for that number whose sum equals the coefficient of the middle term.
Now rewrite the polynomial by splitting the middle term using the two factors that found before and then add the first two terms and last two terms, taking common factors outside from the first and last two terms. Add the four terms of the above step we get the desired factorization.
After factoring the expression solve for $x$, we get the solution.

Complete step-by-step solution:
In this question, we are going to factor the given expression and then find the values of $x$.
First rewrite the given expression in the quadratic form and mark it as$\left( 1 \right)$.
$ \Rightarrow {x^2} - 20x + 100 = 0....\left( 1 \right)$
The given expression is of the quadratic form $a{x^2} + bx + c = 0$
Here the first term is ${x^2}$ and its coefficient is $1$
The middle term is $ - 20x$ and its coefficient is $ - 20$
The last term is $100$ and it is a constant.
First we are going to multiply the coefficient of first term by the last term.
That is, $1 \times 100 = 100$
Next we are going to find factors of $100$ whose sum is equal to $ - 20$
$ \Rightarrow - 10 - 10 = - 20$
By splitting the middle term using the factors $ - 10$ and $ - 10$ in the given expression
$ \Rightarrow {x^2} - 10x - 10x + 100 = 0$
Taking common terms from the first two terms and the last two terms
$ \Rightarrow x\left( {x - 10} \right) - 10\left( {x - 10} \right) = 0$
Taking common factors outside from the two pairs
$ \Rightarrow \left( {x - 10} \right)\left( {x - 10} \right) = 0$
Equate the two factors equal to zero
$ \Rightarrow \left( {x - 10} \right) = 0,\left( {x - 10} \right) = 0$
$ \Rightarrow x = 10,x = 10$
Thus we get the value of $x$ is equal to $10$.

Therefore the required factors of the expression are $\left( {x - 10} \right)\left( {x - 10} \right)$

Note: We can check our factoring by multiplying them all out to see if we get the original expression. If we do, our factoring is correct, otherwise we had to try again.
The following are some of the factoring methods to solve the expression: factoring out the GCF, the sum product pattern, the grouping method, the perfect square trinomial pattern, the difference of square pattern.