Problem 21 - Algebra A Combined Function

Question

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ (m+2)(2 m-6)=5(m-1)-12 $$

Step-by-Step Solution

Answer

The solutions are \(m = 2.5\) and \(m = 1\).

Step 1: Expand the Equation

The first step is to expand both sides of the equation. Start with the left side: \((m+2)(2m-6) = 2m^2 - 6m + 4m - 12 = 2m^2 - 2m - 12\). Now expand the right side: \(5(m-1) - 12 = 5m - 5 - 12 = 5m - 17\). Therefore, the equation becomes \(2m^2 - 2m - 12 = 5m - 17\).

Step 2: Rearrange into a Standard Quadratic Equation

We need to arrange the equation into the standard form \(ax^2 + bx + c = 0\). Move all terms to one side: \(2m^2 - 2m - 12 - 5m + 17 = 0\) which simplifies to \(2m^2 - 7m + 5 = 0\). The equation is now in the standard form where \(a = 2\), \(b = -7\), and \(c = 5\).

Step 3: Apply the Quadratic Formula

We use the quadratic formula, which is \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).Substitute \(a = 2\), \(b = -7\), and \(c = 5\) into the formula: \(m = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \times 2 \times 5}}{2 \times 2}\). Simplifying, we get \(m = \frac{7 \pm \sqrt{49 - 40}}{4}\).

Step 4: Simplify Under the Square Root

Calculate the expression under the square root: \(49 - 40 = 9\). So, \(m = \frac{7 \pm \sqrt{9}}{4} = \frac{7 \pm 3}{4}\).

Step 5: Solve for m

Now we calculate the two possible values for \(m\):1. \(m = \frac{7 + 3}{4} = \frac{10}{4} = 2.5\)2. \(m = \frac{7 - 3}{4} = \frac{4}{4} = 1\) Thus, the solutions are \(m = 2.5\) and \(m = 1\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to find the solutions of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The standard formula to solve these equations is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation. This formula provides a way to find the roots of the equation by calculating the values of \( x \) that solve the equation. It's important to understand each component's role:

• \( -b \): This flips the sign of the coefficient \( b \).
• \( \pm \sqrt{b^2 - 4ac} \): Indicates that there can be two possible solutions, adding and subtracting the square root value.
• \( 2a \): Divides the entire expression, balancing the equation.

Remember that the quadratic formula is applicable when you have correctly identified and inputted the values of \( a \), \( b \), and \( c \). Each value impacts the final solution and must be determined accurately from the equation.

Real Number Solutions

In quadratic equations, real number solutions are the values of the variable that satisfy the equation, specifically when the discriminant is non-negative. The discriminant, \( \Delta \), is the part of the quadratic formula under the square root sign: \( b^2 - 4ac \). Here’s how it affects solution types:

• If \( \Delta > 0 \), there are two distinct real solutions. This means the parabola intersects the x-axis at two points.
• If \( \Delta = 0 \), there is exactly one real solution. The parabola touches the x-axis at a single point, indicating a perfect square trinomial.
• If \( \Delta < 0 \), there are no real solutions. Instead, the solutions are complex or imaginary as the square root of a negative number is not a real number.

In our given problem, the discriminant calculated was \( 9 \), which is positive. This indicates the presence of two real and distinct solutions, meaning we can expect two possible values for \( m \). Knowing whether a quadratic equation has real solutions helps in understanding the nature of the roots without graphing the equation.

Solving Quadratic Equations

Solving quadratic equations involves a series of structured steps to find solutions for the unknown variable. Here's how to effectively solve them:First, ensure that the quadratic equation is written in its standard form \( ax^2 + bx + c = 0 \). This format allows you to identify the coefficients \( a \), \( b \), and \( c \). In our example, through expansion and rearrangement, we found that \( 2m^2 - 7m + 5 = 0 \).The next step is applying the quadratic formula, which will yield exact solutions. Accurate substitution into the formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is crucial. We carefully plugged in the values of \( a = 2 \), \( b = -7 \), and \( c = 5 \) and solved for \( m \).After solving the equation, simplify each part, especially any radical expressions. In our case, we simplified \( \sqrt{9} \) to get exact values of \( m \), which were \( 2.5 \) and \( 1 \). These solutions indicate where the parabola intersects the x-axis.Understanding each step ensures that you can tackle quadratic equations with confidence. This systematic approach also confirms that your solutions are correct and applicable in various real-world contexts.