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In $15-23 :$ a. Find the value of the discriminant and determine if the roots of the quadratic equation are $(1)$ rational and unequal, $(2)$ rational and equal, $(3)$ irrational and unequal, or $(4)$ not real numbers. b. Use any method to find the real roots of the equation if they exist. $$2 x^{2}+7 x=0$$

(a) Discriminant $=7^{2}-4(2)(0)=49$The discriminant is a perfect square number so the roots of the equation are: (1) rational and unequal(b) The roots are $x=0, \frac{7}{2}$

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 3

The Discriminant

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

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so on this problem, we're gonna use discriminative first to you find out more information about this quadratic equation. So the discriminative going to B B squared minus four e c. Which is like the middle off, the scary in the quadratic equation. Our car drug formula. Ah, solution. So if the experiments created and zero, we know that there's two real roots. Um, so let's just let's just all for it right now. And if we know that is equal to zero, we know there's a repeated route. We know that it's less than zero. There's gonna be imaginary bits. So, uh, b squared here is 49 minus ah four times a, which is two times C, which is zero. And that's equal to 49 because 49 is, ah, greater than zero. We know that there's two real roots. We also know that the square root of 49 is equal to seven. So on top of nine of those two routes, there are two riel rational and unequal roots. Probably the best way to solve this would be factory. So if I factor on X term from this equation, I'll get two x plus seven and solving this quarter equation. I get that X can be zero and it can be minus seven over too. And those are my two solutions.

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