Question

How do you factor $10{x^2} - 19x + 7 = 0$ ?

Answer 1

Hint: This equation is the quadratic equation. The general form of the quadratic equation is $a{x^2} + bx + c = 0$. Where ‘a’ is the coefficient of ${x^2}$, ‘b’ is the coefficient of x and ‘c’ is the constant term.
To solve this equation, we will apply the sum-product pattern. During the simplification, we will take out common factors from the two pairs. Then we will rewrite it in factored form.
Therefore, we should follow the below steps:
Apply sum-product patterns.
Make two pairs.
Common factor from two pairs.
Rewrite in factored form.

Complete step-by-step solution:
 Here, the quadratic equation is
$ \Rightarrow 10{x^2} - 19x + 7 = 0$
Let us apply the sum-product pattern in the above equation.
Since the coefficient of ${x^2}$ is 10 and the constant term is 7. Let us multiply 10 and 7. The answer will be 70. We have to find the factors of 70 which sum to -19. Here, the factors are -14 and -5.
Therefore,
$ \Rightarrow 10{x^2} - 14x - 5x + 7 = 0$
Now, make two pairs in the above equation.
$ \Rightarrow \left( {10{x^2} - 14x} \right) - \left( {5x - 7} \right) = 0$
Let us take out the common factor.
$ \Rightarrow 2x\left( {5x - 7} \right) - 1\left( {5x - 7} \right) = 0$
Now, rewrite the above equation in factored form.
$ \Rightarrow \left( {5x - 7} \right)\left( {2x - 1} \right) = 0$
Now,
$ \Rightarrow \left( {5x - 7} \right) = 0$ and $ \Rightarrow \left( {2x - 1} \right) = 0$
Simplify them.
$ \Rightarrow 5x - 7 + 7 = 0 + 7$ and $ \Rightarrow 2x - 1 + 1 = 0 + 1$
That is equal to,
$ \Rightarrow 5x = 7$ and $ \Rightarrow 2x = 1$
Therefore,
$ \Rightarrow x = \dfrac{7}{5}$ and x = \dfrac{1}{2}$

Hence, the roots of the given equation are $\dfrac{7}{5}$ and $\dfrac{1}{2}$.

Note: One important thing is, we can always check our work by multiplying our factors back together, and check that we have got back the original answer.
$ \Rightarrow x = \dfrac{7}{5}$ and $ \Rightarrow x = \dfrac{1}{2}$
Simplify them.
$ \Rightarrow x \times 5 = \dfrac{7}{5} \times 5$ and $ \Rightarrow x \times 2 = \dfrac{1}{2} \times 2$
$ \Rightarrow 5x = 7$ and $ \Rightarrow 2x = 1$
 That is equal to,
$ \Rightarrow 5x - 7 = 0$ and $ \Rightarrow 2x - 1 = 0$
To check our factorization, multiplication goes like this:
$ \Rightarrow \left( {5x - 7} \right)\left( {2x - 1} \right) = 0$
Let us apply multiplication to remove brackets.
$ \Rightarrow 10{x^2} - 14x - 5x + 7 = 0$
Let us simplify it. We will get,
$ \Rightarrow 10{x^2} - 19x + 7 = 0$
Hence, we get our quadratic equation back by applying multiplication.
Here is a list of methods to solve quadratic equations:
Factorization
Completing the square
Using graph
Quadratic formula