Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the quadratic equation by completing the square. Verify your answer graphically. $$4 x^{2}-16 x-5=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(4x^{2}-16x-5=0\) are \(x = 4.29\) and \(x = -0.29\) (approximated to two decimal places).

Step 1: Arrange the Equation

Start by arranging the given equation into a form suitable for completing the square.\n\nDivide the given equation by the coefficient of the squared term to make the coefficient of \(x^2\) equal to 1: \(\frac{4x^{2}-16x-5}{4}\) which simplifies to \(x^2 - 4x - 1.25 = 0\)

Step 2: Complete the Square

The next step involves adding the square of half the coefficient of x to both sides. This will create a perfect square on the left side of the equation.\n\nThe coefficient of x is -4, so half of it is -2, and its square is 4.\n\nAdding 4 to both sides gives: \(x^2 - 4x + 4 = 1.25 + 4\)\n\nTherefore, \(x^2 - 4x + 4 = 5.25\)

Step 3: Factor and Solve for x

Factor the left side of the equation as \((x - 2)^2\), and simplify the right side.\n\nSo, \((x - 2)^2 = 5.25\)\n\nTaking square roots of both sides gives, \(x - 2 = \pm \sqrt{5.25}\)\n\nAdding 2 to both sides gives the roots of the equation: \(x = 2 \pm \sqrt{5.25}\). Therefore the roots are \(x = 4.29\) and \(x = -0.29\) (approximated to two decimal places).

Step 4: Verify Graphically

To verify the answer graphically, plot the function \(y=4x^{2}-16x-5\). The x-axis intercepts, also known as zeros or roots, should be the same as the solution to the equation. From the graph, you will be able to see the intersection point on the x-axis at positions approximately \(x = 4.29\) and \(x = -0.29\), thus confirming the solutions.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial. This technique not only helps in solving equations but also provides a way to derive the quadratic formula. Here's a simplified explanation of the process:

• Start by moving the constant term to the other side of the equation, if necessary. You want the equation in the form of \(ax^2 + bx + c = 0\).
• Ensure the coefficient of \(x^2\) is 1. If it isn't, divide the whole equation by the coefficient.
• Take half the coefficient of the \(x\)-term, square it, and add it to both sides of the equation. This step completes the square.
• The equation now becomes \((x - d)^2 = e\), where \(d\) is half the coefficient of \(x\).
• Solve for \(x\) by taking the square root of both sides and adding or subtracting \(d\) to find the roots.

This approach simplifies solving quadratics and makes understanding their graphing much easier.

Factoring Quadratic Expressions

Factoring quadratic expressions is another common method used to solve quadratic equations. It involves expressing the quadratic as a product of two binomials. Here's how it works:

• Write the quadratic expression in standard form \(ax^2 + bx + c = 0\).
• Find two numbers that multiply to \(a \times c\) and add to \(b\).
• Rewrite the middle term using the two numbers found, splitting it into two separate terms.
• Factor by grouping. This step involves grouping the terms into pairs and factoring out the common factor from each pair.
• The expression should now be in the form of \((px + q)(rx + s) = 0\), and you solve for \(x\) by setting each binomial equal to zero.

Although not always possible for every quadratic equation, when it works, factoring provides the integer solutions directly and is very satisfying.

Verifying Solutions Graphically

Graphical verification of solutions is a visual way to confirm the roots of a quadratic equation. This technique allows you to see where the equation's graph intersects the x-axis, which represents the solutions of the equation.

• First, rewrite the quadratic equation in the form \(y = ax^2 + bx + c\).
• Plot this equation on a graph.
• The graph's intersection points with the x-axis are the roots of the equation.
• These points should match the roots you've calculated algebraically.

Using graphing technology or plotting by hand provides a clear and immediate understanding of the solutions, confirming their accuracy and making the abstract more concrete.

Solving Quadratic Equations

Solving quadratic equations can be approached in several ways, each with its own applications and advantages. The most popular methods include:

• Factoring: Works best when the quadratic can easily be rewritten as a product of binomials. Provides exact roots.
• Completing the Square: Converts any quadratic equation into an easily solvable form. It's particularly useful for deriving the quadratic formula.
• Quadratic Formula: A universal method given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), applicable to any quadratic equation.
• Graphing: Offers visual insight and verification of solutions, showing the parabola's interaction with the x-axis.

Choosing the right method depends on the specific equation and the context in which you're solving it. Understanding each technique enhances your ability to tackle a wide range of quadratic problems effectively.