Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the quadratic equation by completing the square. Verify your answer graphically. $$x^{2}+4 x-32=0$$

Step-by-Step Solution

Answer

The roots of the quadratic equation are \(x = -8\) and \(x = 4\).

Step 1: Rearrange the equation

Rearrange the equation to isolate the terms with the variable on one side: \(x^{2}+4x =32\).

Step 2: Completing the Square

Add the square of half the coefficient of \(x\), which is \((4/2)^{2} = 4\), to both sides to complete the square: \((x^2 + 4x +4) = 32 + 4 = 36.\) This results in a perfect square equation: \((x + 2)^{2} = 36\).

Step 3: Square root of both sides

Take the square root of both sides: \(x + 2 = ±6\).

Step 4: Solve for x

Solve for \(x\): \(x = -2 ±6\), which gives \(x = -2 - 6 = -8\) and \(x = -2 + 6 = 4\). These are the roots of the equation.

Step 5: Verify graphically

Plot the equation \(y = x^{2}+4x - 32\). It forms a parabola. Mark points (-8, 0) and (4, 0) on the graph. These correspond to the roots of the equation.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations. It involves transforming the quadratic into a perfect square trinomial, making it simpler to solve. For the equation \(x^2 + 4x - 32 = 0\), we first rearranged it as \(x^2 + 4x = 32\). The technique of completing the square involves adding and subtracting the same number, turning the left side into a perfect square.

• First, identify the coefficient of \(x\), which is 4 in this case.
  
• Divide that by 2, giving 2, and square it to get 4.
  
• Add 4 to both sides of the equation to maintain equality. This gives us \((x^2 + 4x + 4 = 36)\).

Now, you should see \((x+2)^2 = 36\), a simplified form."

Graphical Verification

Graphical verification is a visual method to confirm the solutions of a quadratic equation. By plotting the quadratic function, you can easily see where it intersects the x-axis. For our equation \(x^2 + 4x - 32 = 0\), you plot the function and look for the points where the curve crosses the x-axis.
These intersection points represent the solutions to the equation. In our case, the roots \(x = -8\) and \(x = 4\) indicate where the curve meets the x-axis at (-8, 0) and (4, 0).
All you need is to:

• Plot the function \(y = x^2 + 4x - 32\).
  
• Identify the points where the graph touches the x-axis.

Graphical verification is useful because it gives a quick and intuitive check of your answers.

Solving Quadratics

Solving quadratic equations is a fundamental skill in algebra. Various methods, like factoring, using the quadratic formula, and completing the square, can be employed. In our example \(x^2 + 4x = 32\) was solved by completing the square.
Here's a quick recap on how to solve quadratic equations efficiently:

• Start by trying to factor the equation if possible.
  
• If factoring is difficult, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  
• Completing the square is another method, transforming the equation into a perfect square.

Choosing the right method depends on the specific equation. Understanding these methods allows for greater flexibility in solving various quadratic problems.