Question

How do you solve $ 5{x^2} + 5x - 1 = 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 quadratic formula for the quadratic equation.
The quadratic formula is as below:
 $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ .
Here, $ \sqrt {{b^2} - 4ac} $ is called the discriminant. And it is denoted by $ \Delta $ .
If $ \Delta $ is greater than 0, then we will get two distinct and real roots.
If $ \Delta $ is less than 0, then we will not get real roots. In this case, we will get two complex numbers.
If $ \Delta $ is equal to 0, then we will get two equal real roots.

Complete step by step answer:
Here, the given quadratic equation is
 $ \Rightarrow 5{x^2} + 5x - 1 = 0 $
We want to find the roots.
First, let us compare the above expression with $ a{x^2} + bx + c = 0 $ .
Here, we get the value of ‘a’ is 5, the value of ‘b’ is 5, and the value of ‘c’ is -1.
Now, let us find the discriminant $ \Delta $ .
 $ \Rightarrow \Delta = {b^2} - 4ac $
Let us substitute the values.
 $ \Rightarrow \Delta = {\left( 5 \right)^2} - 4\left( 5 \right)\left( { - 1} \right) $
Simplify it.
 $ \Rightarrow \Delta = 25 + 20 $
Subtract the right-hand side.
 $ \Rightarrow \Delta = 45 $
Here, $ \Delta $ is greater than 0, then we will get two distinct real roots.
Now,
 $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $
Now, put all the values.
 $ \Rightarrow x = \dfrac{{ - \left( 5 \right) \pm \sqrt {45} }}{{2\left( 5 \right)}} $
That is equal to
 $ \Rightarrow x = \dfrac{{ - 5 \pm 3\sqrt 5 i}}{{10}} $
Hence, the two factors are $ \dfrac{{ - 5 + 3\sqrt 5 i}}{{10}} $ and $ \dfrac{{ - 5 - 3\sqrt 5 i}}{{10}} $ .

Note: One important thing is, we can always check our work by multiplying out factors back together, and check that we have got back the original answer.
Here is a list of methods to solve quadratic equations:
• Factorization
• Completing the square
• Using graph
• Quadratic formula