Problem 88 - Algebra for College Students

Question

Find all real or imaginary solutions to each equation. Use the method of your choice. $$q^{2}+6 q-7=0$$

Step-by-Step Solution

Answer

The solutions are \(q = 1\) and \(q = -7\).

Step 1 - Identify the equation type

The given equation is a quadratic equation in the form of \(ax^{2} + bx + c = 0\). For \(q^2 + 6q - 7 = 0\), \(a = 1\), \(b = 6\), and \(c = -7\).

Step 2 - Use the quadratic formula

The quadratic formula finds the solutions to the equation \(ax^2 + bx + c = 0\) and is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b = 6\), and \(c = -7\) into the formula.

Step 3 - Calculate the discriminant

The discriminant \((b^2 - 4ac)\) determines the nature of the roots. Calculate: \(6^2 - 4(1)(-7) = 36 + 28 = 64\).

Step 4 - Find the roots using the discriminant

Substitute the discriminant value back into the quadratic formula: \(q = \frac{-6 \pm \sqrt{64}}{2(1)}\). This simplifies to: \(q = \frac{-6 \pm 8}{2}\).

Step 5 - Solve for the roots

Calculate the two possible values for \(q\): \(q = \frac{-6 + 8}{2} = 1\) and \(q = \frac{-6 - 8}{2} = -7\).

Key Concepts

quadratic formula

The quadratic formula is a reliable method for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). It's a powerful tool because it works for all types of quadratic equations. The formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. This formula provides us with the solutions, or roots, of the quadratic equation by calculating them directly. By plugging in the coefficients \(a\), \(b\), and \(c\) from the given equation, you can find the roots without factoring or graphing. To use the quadratic formula correctly:

• Identify the values of \(a\), \(b\), and \(c\) from your equation.
• Calculate the discriminant \(b^2 - 4ac\).
• Substitute these values into the quadratic formula to solve for \(x\).

This method is straightforward and ensures you'll always find the correct roots of the equation.

discriminant

The discriminant is a key part of the quadratic formula and can be found within it under the square root symbol. It is calculated using the expression \(b^2 - 4ac\). The value of the discriminant tells us important information about the nature of the roots of the quadratic equation. Here's how to interpret it:

• If the discriminant is positive (greater than zero), there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root (or a repeated root).
• If the discriminant is negative, there are two complex roots (not real).

For the equation \(q^2 + 6q - 7 = 0\), the discriminant is \(6^2 - 4(1)(-7) = 64\), which is positive. This means our quadratic equation has two distinct real roots.

roots of a quadratic equation

Roots of a quadratic equation are the values of the variable that satisfy the equation. They are also known as solutions to the quadratic equation. For \(ax^2 + bx + c = 0\), the roots can be found via different methods: factoring, completing the square, graphing, and using the quadratic formula. The quadratic formula provides an easy way to find these roots, ensuring you get both solutions if they exist. Using our example:

• We determined the equation to solve is \(q^2 + 6q - 7 = 0\).

By substituting into the quadratic formula, we solved for \(q\): \[ q = \frac{-6 \pm \sqrt{64}}{2(1)} = \frac{-6 \pm 8}{2} \]. This gives us two roots: \(1\) and \(-7\), which are the solutions that make the original equation true.

solving quadratic equations

Solving quadratic equations involves finding the roots of the equation \(ax^2 + bx + c = 0\). There are four primary methods to solve these equations:

• Factoring: Breaking down the equation into two binomial expressions and setting each equal to zero.
• Completing the Square: Rearranging the equation to form a perfect square trinomial.
• Graphing: Plotting the quadratic equation and finding the points where the curve intersects the x-axis.
• Quadratic Formula: Using the formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

For our equation, \(q^2 + 6q - 7 = 0\), we used the quadratic formula because it efficiently finds the roots. By substituting \(a = 1\), \(b = 6\), and \(c = -7\), we solved for the roots step by step. This method guarantees that we have calculated the accurate solutions for the quadratic equation.