Question

How do you solve $ y = - 2{x^2} + 5x - 1 $ ?

Answer 1

Hint: Here in this question, we have to solve the given equation, the given equation is in the form of a quadratic equation. This is a quadratic equation for the variable x. By using the formula $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ , we can determine the solutions.

Complete step-by-step answer:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factorising or by using the formula $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ . Now we will assume the value of y as zero and consider the given equation. So the equation is written as $ 0 = - 2{x^2} + 5x - 1 $ .
In general, the quadratic equation is represented as $ a{x^2} + bx + c = 0 $ , when we compare the above equation to the general form of equation the values are as follows. a=-2 b=5 and c=-1. Now substituting these values to the formula for obtaining the roots we have
 $ x = \dfrac{{ - 5 \pm \sqrt {{5^2} - 4( - 2)( - 1)} }}{{2( - 2)}} $
On simplifying the terms, we have
 $ \Rightarrow x = \dfrac{{ - 5 \pm \sqrt {25 - 8} }}{{ - 4}} $
Now subtract 8 from 25 we get
 $ \Rightarrow x = \dfrac{{ - 5 \pm \sqrt {17} }}{{ - 4}} $
The number 17 is a prime number and we don’t have a square root for this. So, we carry the square root as it is.
Therefore, we have $ x = \dfrac{{ - 5 + \sqrt {17} }}{{ - 4}} $ or $ x = \dfrac{{ - 5 - \sqrt {17} }}{{ - 4}} $ . We can simplify for further so we get
x=0.2192 or x=2.2808
Therefore, the approximate roots are (0.2192,0) and (2.2808,0)
For this equation we can also plot the two-dimensional equation by the graph we can also determine the value of x and y.
So, the correct answer is x=0.2192 or x=2.2808”.

Note: The quadratic equation can be solved by using the factorisation method and we also find the roots by using the formula $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ . While factorising we use sum product rule, the sum product rule is given as the product factors of the number c is equal to the sum of the factors which satisfies the value of b.