For what values of a does y = ax2 - 8x + 4 have no x-intercepts?

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 82

4장, 문제 82

For what values of a does y = ax² - 8x + 4 have no x-intercepts?

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: \(ax^2 - 8x + 4 = 0\).

To determine the number of x-intercepts, analyze the discriminant of the quadratic equation, which is given by \(\Delta = b^2 - 4ac\).

Identify the coefficients: here, \(a = a\) (the parameter we want to find), \(b = -8\), and \(c = 4\). Substitute these into the discriminant formula: \(\Delta = (-8)^2 - 4 \cdot a \cdot 4\).

Simplify the discriminant expression: \(\Delta = 64 - 16a\).

For the quadratic to have no x-intercepts, the discriminant must be less than zero: \(64 - 16a < 0\). Solve this inequality for \(a\) to find the values of \(a\) that make the parabola not cross the x-axis.

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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. Understanding the shape and position of the parabola helps analyze its intercepts and other properties.

X-Intercepts of a Quadratic Function

X-intercepts are points where the graph crosses the x-axis, found by setting y = 0 and solving the quadratic equation ax^2 + bx + c = 0. The number of x-intercepts depends on the nature of the roots of this equation, which can be two, one, or none.

Discriminant and Nature of Roots

The discriminant, given by Δ = b^2 - 4ac, determines the number and type of roots of a quadratic equation. If Δ > 0, there are two distinct real roots; if Δ = 0, one real root; and if Δ < 0, no real roots, meaning the parabola does not cross the x-axis.