The discriminant of a quadratic equation is given by
B^2−4ac
Explain in your own words why a quadratic equation has...
...two real solutions if the discriminant is positive
...one real solution if the discriminant is zero
...no real solutions if the discriminant is negative
[Hint: think about the quadratic formula]
But make it look like a 10th-grader did it
The discriminant of a quadratic equation is found by using the formula B^2 - 4ac. Let's break it down into simpler terms.
When we have a quadratic equation, it usually looks something like ax^2 + bx + c = 0. The discriminant helps us determine how many solutions the equation has.
If the discriminant is positive, it means that the expression B^2 - 4ac is greater than zero. When that happens, the quadratic equation will have two real solutions. This is because when we use the quadratic formula (-b ± √(B^2 - 4ac))/2a, we will get two different values for x that make the equation true.
If the discriminant is zero, it means that the expression B^2 - 4ac equals to zero. In this case, the quadratic equation will have only one real solution. When using the quadratic formula, we will get the same value for x twice. It happens when the quadratic equation has a perfect square in it.
Lastly, if the discriminant is negative, it means that the expression B^2 - 4ac is less than zero. When that occurs, the quadratic equation will have no real solutions. We will not find any real values for x when we try to solve the equation using the quadratic formula. This happens when the quadratic equation cannot be factored into real numbers.
In summary, the discriminant helps us determine the number of real solutions for a quadratic equation. Positive discriminant means two real solutions, zero discriminant means one real solution, and negative discriminant means no real solutions.
Hey there! So, when we're talkin' about quadratic equations, we use this thing called the discriminant to sort out the number of real solutions. The discriminant is found by takin' the value of B^2 minus 4 times A times C. Now, let me break it down for you:
1. If the discriminant is positive, well, that's pretty cool! It means that the equation has two real solutions. In other words, we can find two different real numbers that'll solve the equation. Kinda like hittin' two birds with one stone, ya know?
2. Now, if the discriminant is zero, that means it's equal to zilch! And when that happens, our quadratic equation only has one real solution. It's like finding that one special key that'll unlock the door. Just one real number will solve the equation in this case.
3. And lastly, if the discriminant is negative, well, that's a bummer. It means our quadratic equation doesn't have any real solutions at all. It's like searchin' high and low for a unicorn but finding none. No real numbers can solve the equation in this scenario.
So, to sum it up, the discriminant tells us how many real solutions a quadratic equation can have. Positive means 2 real solutions, zero means 1 real solution, and negative means no real solutions at all. Hope that clears things up for you!
Sure, I'll explain it like a 10th-grader.
So, imagine you have a quadratic equation, which is basically an equation that has a variable squared in it. It looks something like this: ax^2 + bx + c = 0. The discriminant is a special thing that helps us figure out how many solutions, or answers, this equation has. It's calculated using this formula: B^2−4ac, where B is the coefficient of x, and a and c are other numbers in the equation.
Now, let's talk about what happens when we calculate this discriminant.
If the discriminant is positive, it means that B^2−4ac is a positive number. In this case, the quadratic equation has two real solutions. Basically, it means that the equation crosses the x-axis at two different points. Imagine that the graph of the equation is like a U shape. So, if the discriminant is positive, the U shape will cross the x-axis at two different places, giving us two real solutions.
If the discriminant is zero, it means that B^2−4ac is equal to zero. In this case, the quadratic equation has one real solution. The U shape of the graph just touches the x-axis at only one point. So, if the discriminant is zero, it means we have only one real solution for the quadratic equation.
Finally, if the discriminant is negative, it means that B^2−4ac is a negative number. And here's the thing: you can't have a square root of a negative number that gives you a real solution. So, if the discriminant is negative, it means the quadratic equation has no real solutions. In terms of the U shape graph, it means the U shape doesn't cross or touch the x-axis at all.
In summary, the discriminant helps us understand how many real solutions a quadratic equation has. Positive discriminant means two real solutions, zero discriminant means one real solution, and negative discriminant means no real solutions.