Not correct.

We never assume the statement we want to prove. That is known as circular logic.

What it should be is :

x + ½ = 2x² + 5x

2x² + 5x - x - ½ = 0

2x² + 4x - ½ = 0

Divide both sides by 2,

x² + 2x - ¼ = 0

Next, check the discriminant ('b² - 4ac')

Discriminant = 2² - 4(1)(-¼)

= 4 + 1

= 5

Since 5 > 0, discriminant > 0.

This means that the equation has 2 real and distinct roots and therefore the curve y = 2x² + 5x intersects the line y = x + ½ at two distinct points.

Thk a lot

By the way, you have gotten your b and c mixed up.

Always write the equation in the form ax² + bx + c = 0

So depending on which side you bring the terms over to , you could have :

2x² + 4x - ½ = 0 →simplified to x² + 2x - ¼ = 0 ①

Or

-2x² - 4x + ½ = 0 → simplified to -x² - 2x + ¼ = 0 ②

You will get from one equation to the other when you multiply by - 1.

In the case of ②,

Discriminant = (-2)² - 4(-1)(¼)

= 4 + 1
= 5

Regardless of starting from ① or ②, the result is the same.

Thank for the advice

No problem. Do revise the quadratic formula and how to interpret it when you have time.

The whole idea of 2 distinct real roots, 1 equal real root and no real roots is based on the square root in that formula.