91Ó°ÊÓ

The discriminant is a key concept in solving quadratic equations. It helps determine the number and type of solutions a quadratic equation has. The discriminant is part of the quadratic formula and is represented as \( b^2 - 4ac \).

For quadratic equations of the form \( ax^2 + bx + c = 0 \):

• If the discriminant is positive (\( > 0 \)), the quadratic equation has two distinct real solutions.
• If the discriminant is zero (\( = 0 \)), the quadratic equation has exactly one real solution. This solution is also a repeated or double root.
• If the discriminant is negative (\( < 0 \)), the quadratic equation has two complex solutions.

In this exercise, to ensure that the quadratic equation \( y = x^2 - 10x + c \) has exactly one x-intercept, we must set the discriminant to zero. Hence, we solve for c when \( b^2 - 4ac = 0 \).

An x-intercept is where the graph of a function crosses the x-axis. For a quadratic equation \( y = ax^2 + bx + c \), the x-intercepts are the solutions for \( x \) when \( y = 0 \).

This means solving \( ax^2 + bx + c = 0 \).

The solutions can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

You can see that the discriminant (\( b^2 - 4ac \)) affects the number of x-intercepts:

• If the discriminant is positive, there are two x-intercepts.
• If the discriminant is zero, there is exactly one x-intercept.
• If the discriminant is negative, the quadratic has no real x-intercepts (they are complex).

In our example, to find \( c \) such that the quadratic equation has exactly one x-intercept, we set the discriminant to zero and solve for c. This means the equation will have a vertex touching the x-axis.

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). There are various methods to solve for \( x \):

• Factoring: Writing the quadratic equation as a product of binomials and solving for \( x \).
• Completing the square: Rewriting the equation in the form \( (x - p)^2 = q \) to find the values of \( x \).
• Using the quadratic formula: Applying the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to directly solve for \( x \).

In our specific problem, we use the discriminant method to determine the value of \( c \) for which the quadratic equation has exactly one x-intercept.

We start with setting the discriminant equal to zero: \( b^2 - 4ac = 0 \). For the given quadratic \( y = x^2 - 10x + c \), substitute \( a = 1 \) and \( b = -10 \) into the discriminant formula:

\( (-10)^2 - 4(1)(c) = 0 \).

Simplify and solve for \( c \):

\( 100 - 4c = 0 \)
\( 4c = 100 \)
\( c = 25 \).

So, \( c \) must be 25 for the quadratic equation to have exactly one x-intercept.