Problem 160 - Algebra and Trigonometry

Question

What is the discriminant and what information does it provide about a quadratic equation?

Step-by-Step Solution

Answer

The discriminant of a quadratic equation is given by \( b^2 - 4ac \). If it's positive, the equation has two distinct real roots. If it's zero, there's one real root. If it's negative, there are no real roots but two complex roots.

Step 1 Define the concept of a discriminant

The discriminant in a quadratic equation is a part of the quadratic formula found under the square root sign: \( b^2 - 4ac \). This part of the equation determines the number and type of solutions.

Step 2 Explain the possible values of discriminants

The value of the discriminant can be positive, zero, or negative. Each of these situations provides different information about a quadratic equation.

Step 3 Detail what it means for the discriminant to be positive

When the discriminant is positive, it indicates that the quadratic equation has two distinct real solutions or roots. This is because the discriminant tells us that we will be able to subtract a real number from another real number in the square root in the quadratic formula. So, a positive discriminant corresponds to two x-intercepts on the graph of the equation.

Step 4 Detail what it means for the discriminant to be zero

When the discriminant is zero, it indicates that the quadratic equation has exactly one real solution or root. This is because we will be subtracting zero in the square root in the quadratic formula. A discriminant of zero corresponds to one x-intercept (also known as a double root) on the graph of the equation.

Step 5 Detail what it means for the discriminant be negative

When the discriminant is negative, it indicates that the quadratic equation has no real solutions or roots, but two complex solutions. This is because we will be taking the square root of a negative number in the quadratic formula which results in imaginary numbers. A negative discriminant corresponds to no x-intercepts on the graph of the equation.