Problem 61 - Algebra for College Students

Question

Use the method of completing the square to solve \(a x^{2}+\) \(b x+c=0\) for \(x\), where \(a, b\), and \(c\) are real numbers and \(a \neq 0\).

Step-by-Step Solution

Answer

Solve for \(x\) by isolating it as \( x = -\frac{b}{2a} \pm \sqrt{\left( \frac{b}{2a} \right)^2 - \frac{c}{a}} \).

Step 1: Standard Form of Quadratic Equation

To solve a quadratic equation using the method of completing the square, start from the standard form of the quadratic: \( a x^2 + b x + c = 0 \). Assume \( a eq 0 \), making it a valid quadratic equation.

Step 2: Divide by Leading Coefficient

Divide every term in the equation by \( a \) to normalize the coefficient of \( x^2 \) to 1, resulting in \( x^2 + \frac{b}{a} x + \frac{c}{a} = 0 \).

Step 3: Move Constant to Right Side

Rearrange the equation by moving the constant term to the right side: \( x^2 + \frac{b}{a} x = -\frac{c}{a} \).

Step 4: Completing the Square

To complete the square, add and subtract \( \left(\frac{b}{2a}\right)^2 \) on the left side of the equation, giving \( x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 = \left( \frac{b}{2a} \right)^2 - \frac{c}{a} \).

Step 5: Factor the Perfect Square Trinomial

Recognize that \( x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 \) is a perfect square trinomial, which factors as \( \left( x + \frac{b}{2a} \right)^2 \). The equation now becomes \( \left( x + \frac{b}{2a} \right)^2 = \left( \frac{b}{2a} \right)^2 - \frac{c}{a} \).

Step 6: Solve for \(x\) by Taking Square Roots

Take the square root of both sides: \( x + \frac{b}{2a} = \pm \sqrt{\left( \frac{b}{2a} \right)^2 - \frac{c}{a}} \).

Step 7: Isolate \(x\)

Solve for \(x\) by isolating it: \( x = -\frac{b}{2a} \pm \sqrt{\left( \frac{b}{2a} \right)^2 - \frac{c}{a}} \).

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The presence of the \(x^2\) term makes it a quadratic equation. It represents a parabola when graphed on a coordinate plane.
Key elements to remember about quadratic equations:

• They always contain the \(x^2\) term.
• They can have either zero, one, or two real roots.
• The graph of a quadratic equation is a parabola, opening upwards if \(a > 0\) and downwards if \(a < 0\).

Solving these equations requires techniques such as factoring, using the quadratic formula, graphing, or completing the square. Each method can provide the roots or solutions of the equation, which are the values of \(x\) that satisfy the equation when substituted back into it.

Standard Form

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). This form is crucial because it provides a consistent structure for solving quadratic equations. It allows us to apply various solving methods systematically.
Why is the standard form important?

• It helps in identifying the coefficients \(a, b,\) and \(c\), which are fundamental in applying formulas or methods like completing the square.
• It serves as a starting point for solving the equation through various algebraic methods.
• It aids in comparing and analyzing quadratic equations by providing a uniform structure.

When dealing with problems, always first ensure the quadratic equation is expressed in this standard form for ease of manipulation and solution.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. For example, \(x^2 + 2hx + h^2\) can be expressed as \((x + h)^2\). Recognizing this pattern is key in the method of completing the square.
How do you complete the square?

• Take the coefficient of \(x\), divide it by 2, and square it to find the term required to complete the square.
• Adjust the equation by adding and subtracting this term, enabling the expression to form a perfect square trinomial.
• Factoring this trinomial simplifies the problem into a recognizable binomial square.

Understanding perfect square trinomials is vital for solving quadratics by completing the square, helping to transform the equation into a solvable form.

Solving Equations

Solving equations refers to the process of finding the values of variables that satisfy the equation. For quadratic equations, several methods exist, and completing the square is a reliable method that is particularly effective when equations are not easily factorable.
Steps to solve by completing the square:

• First, ensure the quadratic equation is in standard form and divide through by \(a\) to normalize the \(x^2\) coefficient.
• Move the constant term to the other side of the equation.
• Add the square of half the coefficient of \(x\) to both sides, creating a perfect square trinomial.
• Factor the trinomial into a binomial square.
• Solve for \(x\) by taking the square root of both sides and isolating \(x\).

Completing the square not only helps in solving the equation but also provides insight into the properties of the quadratic function, such as its vertex. This method, step-by-step, ensures you find the precise roots of the quadratic equation.