Problem 62 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts of the graph of the function. $$y=-x^{2}-4 x+2$$

Step-by-Step Solution

Answer

The x-intercepts for the given function are \(x=2-\sqrt{6}\) and \(x=2+\sqrt{6}\)

Step 1: Write down the equation

The given equation is \(y=-x^{2}-4x+2\). The x-intercepts are the roots of the equation, which means the values for which \(y=0\). Hence re-write the equation as: \(-x^{2}-4x+2 = 0\)

Step 2: Factorise the quadratic equation

This is a quadratic equation in standard form \(-x^2-4x+2 = 0\) where \(a=-1\), \(b=-4\), and \(c=2\) . The quadratic equation can be factorised using the Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)

Step 3: Substitute the coefficients into the formula

Substitute \(a=-1\), \(b=-4\), and \(c=2\) into the Quadratic Formula: \(x = \frac{-(-4) \pm \sqrt{(-4)^{2}-4*(-1)*2}}{2*(-1)}\)

Step 4: Solve the equation

Perform the calculations inside the formula: \(x = \frac{4 \pm \sqrt{16+8}}{-2}\). This simplifies to: \(x = \frac{4 \pm \sqrt{24}}{-2}\), which further simplifies to: \(x = \frac{4 \pm 2\sqrt{6}}{-2}\), giving the solution as \(x=2-\sqrt{6}\) and \(x=2+\sqrt{6}\), which are the x-intercepts for the quadratic function.

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x with a general formula of \( ax^2 + bx + c = 0 \) where \( a \) is not equal to zero. The equation describes a parabola on the graph and the x-values where the graph intersects the x-axis, called x-intercepts, are of significant interest. These points are also referred to as the roots or zeros of the equation.

To find the x-intercepts, we set the quadratic equation equal to zero (\( y=0 \) for graphing purposes) and solve for x. There are different methods to solve quadratic equations such as factoring, using the Quadratic Formula, completing the square, or graphing. Understanding how to manipulate these equations is crucial for solving a wide range of problems in mathematics.

Factoring Quadratic Equations

Factoring is one method of solving quadratic equations which involves expressing the equation as a product of its factors. It works best when the equation can be easily decomposed into two binomials. For instance, the quadratic equation \( ax^2 + bx + c = 0 \) is factorable if there exist two numbers that multiply to \( ac \) and add up to \( b \).

When factoring, we look for two numbers that satisfy these conditions and split the middle term accordingly, followed by grouping and factoring each group. If a quadratic equation does not factor easily, we might have to use the Quadratic Formula or another technique to find its roots. Factoring is a fundamental algebraic technique and often the quickest way to find the x-intercepts if possible.

Quadratic Formula

The Quadratic Formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is a powerful method for solving any quadratic equation, regardless of whether it can be factored easily. It's derived from the process of completing the square and provides the roots by substituting the coefficients \( a \) as the x^2 coefficient, \( b \) as the x coefficient, and \( c \) as the constant term.

Underneath the square root, \( b^2 - 4ac \) is known as the discriminant, which determines the nature of the roots. If positive, there are two real and distinct roots; if zero, there is one real root (the parabola touches the x-axis at a single point); and if negative, there are two complex roots, and no x-intercepts on the graph. Always make sure to include both plus and minus solutions unless they're identical.

Roots of a Quadratic

The roots of a quadratic, also called zeros or x-intercepts, are the points where the graph of the quadratic function crosses the x-axis. In other words, they are the solutions to the equation when set equal to zero. The number of roots depends on the discriminant (\( b^2-4ac \)): two distinct roots, one repeated root, or no real roots at all.

These roots represent the solutions to the problem and hold significant meaning in various applications, such as physics for projectile motion, economics for profit maximization, and geometry for optimizing areas. Identifying the roots of a quadratic is not just about finding the x-intercepts on the graph, but also about understanding the implications of these intercepts in problem-solving scenarios.