Find a value of c so that y = x2 - 10x + c has exactly one x-intercept.

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 81

4장, 문제 81

Find a value of c so that y = x² - 10x + c has exactly one x-intercept.

검증된 단계별 안내

Recall that the x-intercepts of a quadratic function \(y = ax^2 + bx + c\) occur where \(y = 0\), so set the equation equal to zero: \(x^2 - 10x + c = 0\).

For the quadratic to have exactly one x-intercept, the equation must have exactly one real solution. This happens when the discriminant is zero. The discriminant \(\Delta\) is given by \(\Delta = b^2 - 4ac\).

Identify the coefficients from the quadratic: \(a = 1\), \(b = -10\), and \(c = c\) (unknown). Substitute these into the discriminant formula: \(\Delta = (-10)^2 - 4(1)(c)\).

Set the discriminant equal to zero to find the value of \(c\) that gives exactly one solution: \(100 - 4c = 0\).

Solve the equation \(100 - 4c = 0\) for \(c\) to find the required value that makes the quadratic have exactly one x-intercept.

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Quadratic Functions and Their Graphs

A quadratic function is a polynomial of degree two, typically written as y = ax^2 + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. The x-intercepts are points where the graph crosses the x-axis, found by solving y = 0.

Discriminant of a Quadratic Equation

The discriminant, given by Δ = b^2 - 4ac, determines the nature of the roots of a quadratic equation ax^2 + bx + c = 0. If Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real root (a repeated root); if Δ < 0, there are no real roots.

Condition for Exactly One x-Intercept

For a quadratic function to have exactly one x-intercept, its graph must touch the x-axis at a single point. This occurs when the discriminant is zero, meaning the quadratic has a repeated root. Setting Δ = 0 allows solving for the value of c that satisfies this condition.