Problem 8 - Introductory Algebra for College Students

Question

Find the \(x\) -intercepts for the parabola whose equation is given. If the \(x\) -intercepts are irrational numbers, round your answers to the nearest tenth. $$y=-x^{2}-2 x+3$$

Step-by-Step Solution

Answer

The x-intercepts of the parabola are \(x = -3\) and \(x = 1\)

Step 1: Identify the coefficients of the quadratic equation

The equation is in the form \(ax^2 + bx + c = 0\). Here, \(a = -1, b = -2, and c = 3\).

Step 2: Substitute the coefficients into the quadratic formula

The quadratic formula is \(-b ± √(b^2 - 4ac) / 2a\). Substituting a, b, and c gives us \(-(-2) ± √((-2)^2 - 4(-1)(3)) / 2(-1)\). This simplifies to \(2 ± √(4 + 12) / -2\). Finally, it results in \(2 ± √16 / -2\).

Step 3: Simplify the equation for the roots of the equation

Simplify under the square root and divide by -2 to get the answer : \(2 ± 4 / -2\). This simplifies to obtain two roots: \(x= -1 ± 2\). These roots correspond to the x-intercepts -1-2=-3 and -1+2=1.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful algebraic expression that allows us to find the roots of any quadratic equation. In the standard form, a quadratic equation looks like this: \( ax^2 + bx + c = 0 \) where \( a \) is the coefficient of the \( x^2 \) term, \( b \) is the coefficient of the \( x \) term, and \( c \) represents the constant. To find the roots, or the values of \( x \) where the equation equals zero, we use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The plus-minus sign \( \pm \) indicates that there will be two solutions for \( x \) because the square root term can be either positive or negative. These roots represent the points at which a parabola crosses the x-axis; in other words, the x-intercepts. It's essential to substitute the coefficients accurately and perform the operations in the correct order to get the right answer.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. Essentially, solving for the variable \( x \) means finding the values that make the equation true. There are several methods to solve quadratics, including factoring, completing the square, graphing, and using the aforementioned quadratic formula. The latter is the most general method and works for any form of quadratic equation. For the given equation \( y = -x^2 - 2x + 3 \), we follow a systematic approach by first identifying the coefficients \( (a, b, c) \) and then substituting them into the quadratic formula. By simplifying the resulting expression, we discover the x-intercepts of the parabola. Understanding each step carefully ensures that students do not make errors in calculation and find the accurate solutions.

Roots of a Quadratic

The roots of a quadratic equation are the solutions to the equation; they are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \) when \( y \) is set to zero. These roots can be real or complex numbers, and in the context of a graph, they represent where the parabola intersects the x-axis. The discriminant \( (b^2 - 4ac) \) within the quadratic formula is a helpful tool in predicting the nature of the roots:

• If the discriminant is positive, there are two real and distinct (different) roots.
• If the discriminant is zero, there is one real root, or two real and identical roots.
• If the discriminant is negative, there are no real roots, but two complex roots.

In the exercise in question, the discriminant is \( 16 \) \( (4 + 12) \) which signals that two real and distinct roots exist. As observed in the solution, the roots are \( -3 \) and \( 1 \) after performing the appropriate arithmetic.

Parabola Analysis

Parabola analysis involves understanding the features of the graph of a quadratic equation. A parabola can open upwards or downwards, depending on the sign of the coefficient \( a \) of the \( x^2 \) term. If \( a \) is positive, it opens upwards, and if \( a \) is negative, as in our exercise \( y=-x^2-2x+3 \) with \( a=-1 \), it opens downwards. The vertex of the parabola is the highest or lowest point on the graph and can be found analytically or through completing the square. Other important points include the axis of symmetry, which is a vertical line that divides the parabola into two mirror images, and the x-intercepts or roots, which we've focused on heavily. By plotting these critical features and considering the parabola's direction, one can get a clear picture of the graph's shape and behavior.