Question

How do you factor the quadratic equation \[8{{z}^{2}}-4z+10z-5\]?

Answer 1

Hint: In this problem, we have to find the factors for the given equation. We can see that the given equation is already in an easier manner as we can split it to form factors. We can first split the given equation into two terms, we can then take common terms outside the bracket. We will get a new format. We can again take the common terms separately to find the factors. We can then get the required factors.

Complete step-by-step solution:
We know that the given expression is,
\[8{{z}^{2}}-4z+10z-5\]
We can now write the first two terms and the last two terms to take common terms separately, we get
\[\Rightarrow \left( 8{{z}^{2}}-4z \right)+\left( 10z-5 \right)\]
Now we can take the common terms outside the brackets respectively. The first term has 4z as a common term and the second term has 2 as a common term, we get
\[\Rightarrow 4z\left( 2z-1 \right)+5\left( 2z-1 \right)\]
We can again take the common factor first then the remaining terms to make a factor, we get
\[\Rightarrow \left( 4z+5 \right)\left( 2z-1 \right)\]
Therefore, the factors are \[\left( 4z+5 \right)\left( 2z-1 \right)\].

Note: Students make mistakes while taking the common terms outside the equation. We should always remember that when the common terms are multiplied again it should give the previous equation. We can also solve for x to find the factor of the given equation using the quadratic formula. We can also check for the factors to be correct by multiplying the two factors to get the given equation.