Question

How do you solve $5{x^2} + 17x = 12$?

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 5{x^2} + 17x = 12$
Let us subtract 12 on both sides.
$ \Rightarrow 5{x^2} + 17x - 12 = 12 - 12$
That is equal to,
$ \Rightarrow 5{x^2} + 17x - 12 = 0$
Let us apply the sum-product pattern in the above equation.
Since the coefficient of ${x^2}$ is 5 and the constant term is -12. Let us multiply 5 and -12. The answer will be -60. We have to find the factors of 60 which sum to 17. Here, the factors are 20 and -3.
Therefore,
$ \Rightarrow 5{x^2} + 20x - 3x - 12 = 0$
Now, make two pairs in the above equation.
$ \Rightarrow \left( {5{x^2} + 20x} \right) - \left( {3x + 12} \right) = 0$
Let us take out the common factor.
$ \Rightarrow 5x\left( {x + 4} \right) - 3\left( {x + 4} \right) = 0$
Now, rewrite the above equation in factored form.
$ \Rightarrow \left( {x + 4} \right)\left( {5x - 3} \right) = 0$
Now,
$ \Rightarrow \left( {5x - 3} \right) = 0$ and $ \Rightarrow \left( {x + 4} \right) = 0$
Simplify them.
$ \Rightarrow 5x - 3 + 3 = 0 + 3$ and $ \Rightarrow x + 4 - 4 = 0 - 4$
That is equal to,
$ \Rightarrow 5x = 3$ and $ \Rightarrow x = - 4$
Therefore,
$ \Rightarrow x = \dfrac{3}{5}$ and $ \Rightarrow x = - 4$

Hence, the roots of the given equation are $\dfrac{3}{5}$ and $-4.$

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{3}{5}$ and $ \Rightarrow x = - 4$
Simplify them.
$ \Rightarrow x \times 5 = \dfrac{3}{5} \times 5$ and $ \Rightarrow x + 4 = - 4 + 4$
$ \Rightarrow 5x = 3$ and $ \Rightarrow x + 4 = 0$
 That is equal to,
$ \Rightarrow 5x - 3 = 0$ and $ \Rightarrow x + 4 = 0$
To check our factorization, multiplication goes like this:
$ \Rightarrow \left( {5x - 3} \right)\left( {x + 4} \right) = 0$
Let us apply multiplication to remove brackets.
$ \Rightarrow 5{x^2} + 20x - 3x - 12 = 0$
Let us simplify it. We will get,
$ \Rightarrow 5{x^2} + 17x - 12 = 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