Problem 44 - Beginning and Intermediate Algebra

Question

Solve each equation. $$g(3 g+11)=70$$

Step-by-Step Solution

Answer

The roots of the quadratic equation \(g(3g + 11) = 70\) are \(g_1 = \frac{10}{3}\) and \(g_2 = -7\).

Step 1: Expand the Expression

First, we need to expand the expression and simplify it. We do this by using the distributive property of multiplication over addition: \(g(3g + 11) = 70 \\ 3g^2 + 11g - 70 = 0\) Now the equation is in standard quadratic form.

Step 2: Write Down the Quadratic Formula

The quadratic formula is used to find the roots of a quadratic equation in the form \(ax^2 + bx + c = 0\). The general formula is given as follows: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) For our equation, we have \(a = 3\), \(b = 11\), and \(c = -70\).

Step 3: Plug in the Values and Solve for g

We now plug the values of \(a\), \(b\), and \(c\) into the quadratic formula and solve for \(g\): \(g = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 3 \cdot (-70)}}{2 \cdot 3}\)

Step 4: Simplify and Calculate the Roots

Let's simplify the equation further and calculate the roots: \(g = \frac{-11 \pm \sqrt{121 + 840}}{6} \\ g = \frac{-11 \pm \sqrt{961}}{6} \\ g = \frac{-11 \pm 31}{6}\) Now we have two possible values for \(g\): \(g_1 = \frac{-11 + 31}{6} = \frac{20}{6} = \frac{10}{3}\) \(g_2 = \frac{-11 - 31}{6} = \frac{-42}{6} = -7\) The roots of the given equation are \(g_1 = \frac{10}{3}\) and \(g_2 = -7\).

Key Concepts

Distributive Property

The distributive property is fundamental in helping us expand expressions, especially when dealing with algebraic equations. It's the mathematical rule that distributes multiplication over addition or subtraction within parenthesis. Essentially, when we apply this property, we multiply each term inside the parenthesis by the term outside.
For example, in our original exercise with the equation \(g(3g + 11) = 70\), the distributive property allows us to write it as:

• Multiply \(g\) by \(3g\) to get \(3g^2\).
• Multiply \(g\) by \(11\) to get \(11g\).

This gives us the expanded quadratic expression: \(3g^2 + 11g - 70 = 0\). By using this property, you can always simplify and restate expressions, which makes it easier to solve them or proceed to use other mathematical tools, such as the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation. A quadratic equation typically looks like this: \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps you solve for \(x\) even when other methods like factoring are difficult or not possible. The "\(\pm\)" in the formula signifies that there could be two possible solutions.
In our example, we have:

• \(a = 3\),
• \(b = 11\),
• \(c = -70\).

By substituting these values into the quadratic formula, you connect the derived quadratic expression to its potential solutions, as demonstrated in our solution step. This formula is essential for understanding and solving quadratic equations, no matter how complex they seem.

Roots of a Quadratic Equation

Finding the roots of a quadratic equation involves determining the values of the variable that make the equation true, meaning it equals zero. These roots or solutions can be real or complex numbers. For real solutions, you might have:

• Two distinct real roots.
• One unique real root (also known as a repeated root).
• No real roots (if the solutions are complex numbers).

In the original problem, we solve it by plugging values into the quadratic formula. After calculations, we find:

• The first root \(g_1 = \frac{10}{3}\).
• The second root \(g_2 = -7\).

The sum, difference, multiplication, and factorization of these roots can explain many properties of the quadratic function. The discriminant \(b^2 - 4ac\), found under the square root in the quadratic formula, also gives insights about the nature of the roots. If \(b^2 - 4ac > 0\), the equation has two real roots; if \(b^2 - 4ac = 0\), there is one repeated real root; and if \(b^2 - 4ac < 0\), the roots are complex. Understanding these roots is essential for solving various real-world problems.