Problem 352 - Elementary Algebra

Question

Solve. \(q^{2}-12 q-13=0\)

Step-by-Step Solution

Answer

The solutions are \(q = 13\) and \(q = -1\).

Step 1: Identify the quadratic equation

The given quadratic equation is \(q^{2} - 12q - 13 = 0\).

Step 2: Write the quadratic equation in standard form

The standard form of a quadratic equation is \(ax^{2} + bx + c = 0\). Here, \(a = 1\), \(b = -12\), and \(c = -13\).

Step 3: Use the quadratic formula

The quadratic formula to solve equations of this form is given by \(q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 4: Substitute the values of a, b, and c into the quadratic formula

Substitute \(a = 1\), \(b = -12\), and \(c = -13\) into the formula: \[ q = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \, (1) \, (-13)}}{2 \, (1)} \].

Step 5: Simplify the expression under the square root

Calculate the value under the square root: \((-12)^2 - 4ac = 144 + 52 = 196\).

Step 6: Calculate the square root

The square root of 196 is \(\sqrt{196} = 14\).

Step 7: Solve for q

Now substitute back into the quadratic formula: \[ q = \frac{12 \pm 14}{2} \]. This will give us two solutions:

Step 8: Find the two possible solutions

1) \( q = \frac{12 + 14}{2} = \frac{26}{2} = 13 \) 2) \( q = \frac{12 - 14}{2} = \frac{-2}{2} = -1 \).

Key Concepts

The Quadratic Formula

To solve quadratic equations, the quadratic formula is a powerful tool. It is given by the equation \[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula works for any quadratic equation, which is any equation of the form \[ ax^2 + bx + c = 0 \]. Here, 'a', 'b', and 'c' are coefficients in the equation.

• 'a' is the coefficient of the quadratic term (i.e., the term with \(x^2\)).
• 'b' is the coefficient of the linear term (i.e., the term with \(x\)).
• 'c' is the constant term.

The quadratic formula not only helps find the values of 'q' that satisfy the equation, but it also provides a complete solution regardless of the complexity of the equation.

Solving Quadratic Equations

Solving quadratic equations involves a few steps to simplify and use the quadratic formula effectively. Here's a clear breakdown:

1. Identify the components of the quadratic equation: Look for the coefficients 'a', 'b', and 'c'.
2. Substitute these coefficients into the quadratic formula.
3. Simplify the expression under the square root, known as the discriminant (\(b^2 - 4ac\)).
4. Calculate the square root of the discriminant.
5. Use the results to find the two possible values for 'q'.

For example, in the equation \(q^{2} - 12q - 13 = 0\), we first identify that \(a = 1\), \(b = -12\), and \(c = -13\). We then substitute these values into the quadratic formula:
\[ q = \frac{12 \pm 14}{2} \].
This gives us the two possible solutions for 'q': 13 and -1.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Writing quadratic equations in this form allows us to easily identify the coefficients and apply the quadratic formula.

• \(a\) is the coefficient of \(x^2\), and it determines the width and direction of the parabola formed by the quadratic function.
• \(b\) is the coefficient of \(x\), affecting the symmetry of the graph.
• \(c\) is the constant term, determining where the graph intersects the y-axis.

For the equation \(q^{2} - 12q - 13 = 0\), it is already in standard form where \(a = 1\), \(b = -12\), and \(c = -13\). This form is a prerequisite for using the quadratic formula and solving the equation effectively.