Problem 46 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. $$ x^{2}+16 x+9=0 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(x^{2}+16 x+9=0\) are \(x1 = -0.58\) and \(x2 = -15.42\).

Step 1: Identify a, b and c

In the equation \(x^{2} + 16x + 9 = 0\), we have \(a = 1\), \(b = 16\), and \(c = 9\). These are the coefficients in the equation.

Step 2: Substitute in the quadratic formula

We can substitute these coefficients into the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). This becomes \(x = \frac{-16 \pm \sqrt{{16}^{2} - 4(1)(9)}}{2(1)}\).

Step 3: Calculate the discriminant

Next, we perform the calculations inside the square root, which is also known as the discriminant: \(b^{2} - 4ac = {16}^{2} - 4(1)(9) = 256 - 36 = 220\).

Step 4: Simplify the square root

We continue to simplify inside the square root, \(\sqrt{220}\) which gives us around \(14.8324\).

Step 5: Solve for x

We can now substitute \(\sqrt{220}\) into the equation from Step 2, which gives two possible solutions, \(x = \frac{-16 + 14.8324}{2}\) and \(x = \frac{-16 - 14.8324}{2}\). The solutions to these operations give us \(x1 = -0.5838\) and \(x2 = -15.4162\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra used to find the solutions of quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula is defined as follows: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows us to find the values of \(x\) where the quadratic equation equals zero. This step-by-step approach involves determining the coefficients \(a\), \(b\), and \(c\) from the equation and substituting them directly into the formula.

• **\(a\)** is the coefficient of \(x^2\),
• **\(b\)** is the coefficient of \(x\), and
  
• **\(c\)** is the constant term.

The quadratic formula simplifies finding roots by offering a direct calculation method without factoring. This general approach works for any quadratic equation, providing that the discriminant (the part under the square root sign) is non-negative.

Discriminant

The discriminant is a crucial part of the quadratic formula, represented by \(b^2 - 4ac\). It helps determine the nature of the roots of the equation. Calculating the discriminant is a vital step to understanding the number and type of solutions a quadratic equation may have.

• If the discriminant is **positive**, the quadratic equation has **two distinct real solutions**.
• If it is **zero**, there is exactly **one real solution** (or a repeated root).
• A **negative** discriminant indicates there are **no real solutions**, only complex ones.

For instance, in the given equation \(x^2 + 16x + 9 = 0\), the discriminant is calculated as \(256 - 36 = 220\), which is positive. This signals that the equation has two distinct real solutions.

Solutions of Quadratic Equations

The solutions to a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). When using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
There are typically two solutions, which we derive by evaluating the expression under and following the square root. The "\( \pm \)" symbol indicates that we must compute two separate values, known as the roots of the equation. For example, with \(b^2 - 4ac = 220\) (from our equation), the square root \( \sqrt{220} \approx 14.8324\), leads to the solutions:

• \(x_1 = \frac{-16 + 14.8324}{2} \approx -0.5838\),
• \(x_2 = \frac{-16 - 14.8324}{2} \approx -15.4162\).

These values are the points where the quadratic equation touches or crosses the \(x\)-axis on a graph. Quadratic equations hence can have two, one, or no real solutions, depending on the discriminant value.