Problem 13 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$x^{2}-8 x-9=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 9\) and \(x = -1\).

Step 1: Identify the Equation Type

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -8\), and \(c = -9\). Our task is to solve it by factoring.

Step 2: Set up the Factoring Problem

To factor the quadratic expression \(x^2 - 8x - 9\), we need two numbers whose product is the constant term \(-9\) and whose sum is the linear coefficient \(-8\). Identify these two numbers.

Step 3: Find the Factor Pair

The factor pair of \(-9\) that adds up to \(-8\) is \(-9\) and \(1\). We can verify this because \(-9 + 1 = -8\) and \(-9 \times 1 = -9\).

Step 4: Write the Factored Form

Use the identified numbers to write the factored form of the quadratic: \((x - 9)(x + 1) = 0\). This shows the expression factored into two binomials.

Step 5: Solve for the Roots

Set each binomial equal to zero: \(x - 9 = 0\) and \(x + 1 = 0\). Solving these gives \(x = 9\) and \(x = -1\).

Step 6: Verify the Solution

Check the solutions by substituting back into the original equation. For \(x = 9\): \(9^2 - 8 \times 9 - 9 = 0\), and for \(x = -1\): \((-1)^2 - 8 \times (-1) - 9 = 0\). Both satisfy the original equation.

Key Concepts

Factoring

Factoring is a method used to solve quadratic equations by expressing a quadratic trinomial as a product of two binomials. It involves finding two numbers that serve specific purposes: they need to multiply to give the constant term (the term without an \(x\)) and add up to the coefficient of the linear term. For the quadratic equation \(x^2 - 8x - 9 = 0\), we identified our constants where \(a = 1\), \(b = -8\), and \(c = -9\).
The trick to factoring is to break the middle term (\(b = -8\)) into two terms that relate to the factors chosen. For example, in this equation, we choose \(-9\) and \(1\) because their product is \(-9\) (matching our constant term \(c\)) and their sum is \(-8\) (matching our linear term \(b\)). These factors are then used to break the middle term and factor the quadratic into binomials. The factored form is \((x - 9)(x + 1)\). Using this method allows us to easily find the roots of the quadratic equation by setting each binomial to zero.

Roots of Equations

Every quadratic equation holds two solutions known as roots. These roots are where the graph of the equation intersects the x-axis. Finding the roots by factoring involves setting the factored form of the equation equal to zero, such as in our example from \((x - 9)(x + 1) = 0\).
Once the equation is factored, the next step is to solve for each root separately. Here, we solve the two equations given by the factored form:

• \(x - 9 = 0\)
• \(x + 1 = 0\)

This involves basic algebra: adding 9 on both sides of the first equation yields \(x = 9\), and adding -1 to the second equation yields \(x = -1\). Thus, the roots of the equation are \(x = 9\) and \(x = -1\). These roots represent the points on the graph where the parabola crosses the x-axis, providing solutions to the equation.

Verifying Solutions

Verification of solutions is an essential step to ensure accuracy in solving quadratic equations. Once roots are found by factoring, we substitute these values back into the original equation to verify they satisfy it. For the equation \(x^2 - 8x - 9 = 0\), we have roots at \(x = 9\) and \(x = -1\).
Let's check these roots:

• Substituting \(x = 9\): \[9^2 - 8 \times 9 - 9 = 81 - 72 - 9 = 0\]
• Substituting \(x = -1\):\[(-1)^2 - 8 \times (-1) - 9 = 1 + 8 - 9 = 0\]

Both calculations yield zero, confirming that \(x = 9\) and \(x = -1\) are indeed solutions to the original equation. Verifying solutions ensures the correctness of the mathematical processes performed and provides confidence in the results achieved.