Y=4x²-5x+1

To prepare for your university exams, it is important to be familiar with different aspects of quadratic equations. Let's analyze the given equation:

y = 4x² - 5x + 1

This equation is in standard quadratic form, which is given by ax² + bx + c = 0, where a, b, and c are constants.

In this case, a = 4, b = -5, and c = 1.

To further understand and analyze the equation, we can calculate the discriminant (Δ) using the formula:

Δ = b² - 4ac

Δ = (-5)² - 4(4)(1) Δ = 25 - 16 Δ = 9

Since the discriminant is positive (Δ > 0), the equation has two distinct real roots. This means that the graph of the quadratic equation will intersect the x-axis at two different points.

To find the x-intercepts (or roots) of the equation, we can use the quadratic formula:

x = (-b ± √Δ) / (2a)

Plugging in the values we previously found:

x = (-(-5) ± √9) / (2(4)) x = (5 ± √9) / 8

Simplifying further, we get two possible solutions:

x₁ = (5 + √9) / 8 x₂ = (5 - √9) / 8

These are the solutions to the given quadratic equation. Remember to retain both the positive and negative values, as indicated by the "±" symbol, when solving quadratic equations.

Additionally, it is essential to practice graphing quadratic equations to understand their behavior and identify important features such as the vertex, axis of symmetry, and direction of opening.