Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the quadratic equation by completing the square. Verify your answer graphically. $$-x^{2}+6 x-16=0$$

Step-by-Step Solution

Answer

The roots of the quadratic equation \( -x^{2}+6x-16=0 \) are \( x = 3 + \sqrt{7}i \) and \( x = 3 - \sqrt{7}i \)

Step 1: Write the given equation in standard form

The given equation is already in standard form, i.e, \( -x^{2}+6x-16=0 \)

Step 2: Move the constant term to the right side

To complete the square, move constant term on the other side of equation, giving \( -x^{2}+6x = 16 \)

Step 3: Divide the equation by coefficient of x^2

Dividing each side by -1, we get \( x^2-6x = -16 \)

Step 4: Move the coefficient of x to complete the square

Half the coefficient of \( x \) is -3. Square this value, i.e., (-3)² = 9, and add it to both sides, giving \( x^2-6x+9 = -16+9 \),which simplifies to \( x^2-6x+9 = -7 \)

Step 5: Simplify the equation to find the roots

Taking square root on both sides, we get \( (x - 3) = \sqrt{-7} \). As there is a negative sign under the square root, there will be no real roots. Therefore, the roots are complex i.e. \( (x - 3) = \pm \sqrt{7}i \). So, the roots of the equation are \( x = 3 + \sqrt{7}i \) and \( x = 3 - \sqrt{7}i \)

Step 6: Verify the roots graphically

Plot the function \( y = -x^{2}+6x-16 \). Since the roots are complex, they won't intersect the x-axis, which is true as per the graphical representation.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by turning a quadratic expression into a perfect square trinomial. This makes it easier to solve for the variable, especially when factoring isn't straightforward.

To complete the square for the quadratic equation \(-x^2 + 6x - 16 = 0\), we first need to rearrange it into the form \(x^2 - 6x = -16\) by moving the constant to the other side and ensuring the leading coefficient of \(x^2\) is positive.

• Find half of the coefficient of \(x\), which is -3, and then square it to get 9.
• Add 9 to both sides of the equation to maintain equality. This transforms the equation into \(x^2 - 6x + 9 = -7\).

Now, the left side of the equation is a perfect square trinomial, \((x - 3)^2\), which simplifies solving the equation further.

Complex Roots

When solving \((x - 3)^2 = -7\), taking the square root reveals a negative number under the radical. This indicates the presence of complex roots.

The expression becomes \(x - 3 = \pm \sqrt{7}i\). Here, \(i\) is the imaginary unit, defined as \(\sqrt{-1}\). Complex roots occur in conjugate pairs, that is, pairs like \(3 + \sqrt{7}i\) and \(3 - \sqrt{7}i\).

• Complex roots mean the graph of the quadratic doesn't cross the x-axis.
• This makes completing the square particularly useful, as it leads directly to the identification of complex solutions.

Graphical Verification

Graphically verifying the solution involves plotting the quadratic equation \(y = -x^2 + 6x - 16\). When you plot it:

• The curve opens downwards, due to the negative sign in front of \(x^2\).
• Since the roots are complex, the graph does not touch or intersect the x-axis.

Seeing that the parabola doesn't cross the x-axis confirms the absence of real roots and the presence of complex roots. This step ensures that the analytical solutions match the visual insight from the graph.

Standard Form of Quadratic

Quadratic equations are often expressed in standard form as \(ax^2 + bx + c = 0\). This format is crucial for applying various solution methods, like completing the square or the quadratic formula.

In our example, the equation \(-x^2 + 6x - 16 = 0\) is already in standard form. Here:

• \(a = -1\), \(b = 6\), and \(c = -16\).
• Maintaining this form simplifies the process of identifying necessary transformations or calculations.

The consistency of the standard form allows for systematic approaches to finding solutions, whether they are real or complex.