Problem 61 - Algebra 1

Question

Find the x-intercepts of the graph of the equation. $$y=-5 x^{2}+5 x+5$$

Step-by-Step Solution

Answer

The x-intercepts of the graph of the equation are \(x = 1 / 2 - sqrt(5) / 2\) and \(x = 1 / 2 + sqrt(5) / 2\).

Step 1: Set the equation equal to zero

Begin by setting the equation equal to zero, as we want to find the values of x that make y equal to zero. This gives the equation: \(0 = -5x^{2} + 5x + 5\)

Step 2: Simplify the equation

To make the equation easier to manage, divide every term by -5. This gives us: \(0 = x^{2} - x - 1\)

Step 3: Apply the quadratic formula

The quadratic formula is used to find the roots of a quadratic equation. The quadratic formula states that for any quadratic equation of the form \(ax^2 + bx + c\), the solutions for x can be found using the formula:\(x = [-b ± sqrt(b^2 - 4ac)] / 2a\).The solutions for x then are:\(x_1 = [1 + sqrt((-1)^2 - 4*(-1)) / (2*-1) = 1 / 2 - sqrt(5) / 2\)and\(x_2 = [1 - sqrt((-1)^2 - 4*(-1)) / (2*-1) = 1 / 2 + sqrt(5) / 2\)

Key Concepts

Quadratic Formula

One of the most remarkable achievements in algebra is the quadratic formula, which provides a solution for finding the roots of any quadratic equation. A quadratic equation can be recognized by its standard form, which appears as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\).

The quadratic formula is expressed as \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\). This formula implies that by substituting the coefficients \(a\), \(b\), and \(c\) into the formula, one can solve for the two roots (or x-intercepts) of the equation. These roots, which might be real or complex numbers, represent the points at which the graph of the quadratic equation crosses the x-axis.

• The '+' sign corresponds to one of the roots.
• The '−' sign corresponds to the other root.

Understanding and applying the quadratic formula is an essential skill for students tackling algebra, as it gives a methodical approach to solving any quadratic equation.

Finding Roots of Quadratic Equations

The roots of quadratic equations, also known as zeros, solutions, or x-intercepts, are the values of \(x\) where the graph of the equation intersects the x-axis. These roots can be found using various methods, including factoring, completing the square, and, most commonly, the quadratic formula.

When applying the quadratic formula, it's important to pay attention to the discriminant, \(b^2 - 4ac\). This part of the formula determines the nature of the roots:

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root (also called a repeated or double root).
• If the discriminant is negative, the equation has two complex roots.

The quadratic formula is particularly useful because it can be applied to any quadratic equation, regardless of whether it is factorable or not. Remember that simplifying the equation and identifying the coefficients correctly are crucial steps before applying the formula.

Simplifying Quadratic Equations

Simplifying a quadratic equation can make the process of finding its roots much easier. In the context of the quadratic formula and finding x-intercepts, simplifying often means making the leading coefficient equal to 1, which is also known as normalizing the quadratic equation. This involves dividing every term in the equation by the leading coefficient if it's not already 1.

Once normalized, the quadratic equation takes the simpler form \(x^2 + bx' + c' = 0\), where \(b'\) and \(c'\) are the new coefficients obtained after the division. Normalization makes the application of the quadratic formula more straightforward and helps in reducing calculation errors. It is especially effective when the original leading coefficient has a common factor with other terms.

Exercise Improvement Tip:

When dividing each term to simplify a quadratic equation, ensure that you simplify completely and consistently, reducing fractions where possible. This approach not only makes the use of the quadratic formula easier but also helps to avoid mistakes during calculations.