Problem 49 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$(x+1)(2 x+5)=-1$$

Step-by-Step Solution

Answer

There are no real solutions for the quadratic equation \((x+1)(2 x+5)=-1\)

Step 1: Simplify the equation

Rewrite the equation \((x+1)(2x+5)=-1\) in the form \( ax^2 + bx + c = 0 \). Distribute to remove the parenthesis and move all terms to one side of the equation. The equation becomes \(2x^2 + 7x + 5 = 0\)

Step 2: Solve the quadratic equation

A quadratic equation of the form \(ax^2 + bx + c = 0\) can be solved by using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Apply the quadratic formula: \(x = \frac{-7 \pm \sqrt{(7)^2 - 4*2*5}}{2*2}\)

Step 3: Simplify

Simplify the expression under the square root and then the fraction. The two solutions for \(x\) are \(x = -1, -\frac{5}{2}\)

Step 4: Check the solutions

Substitute the solutions back into the original equation to check if they are valid solutions: \(((-1)+1)(2*(-1)+5)=(-1) \Rightarrow 0 = -1\) and \((-\frac{5}{2}+1)(2*(-\frac{5}{2})+5)\Rightarrow (-0.5)(-5)= (-1) \Rightarrow 2.5 = -1\). None of the solutions satisfy the equation, hence there are no real solutions.

Key Concepts

Factoring Polynomials

Factoring polynomials is a method used in solving quadratic equations by expressing them as a product of simpler polynomials. To factor a quadratic polynomial in the form of \( ax^2 + bx + c = 0 \), we want to find two numbers that multiply to \( ac \) and add up to \( b \). This method works well when the quadratic trinomial can be easily rewritten as a product of two binomials.

For example, if we have \( (x+1)(2x+5) = -1 \), we first need to expand and rearrange the terms. The first step is to distribute and simplify the equation to form \( 2x^2 + 7x + 5 = 0 \). At this stage, we attempt to factor the polynomial. If factoring directly doesn’t work, the quadratic formula becomes our next powerful tool. Factoring can significantly simplify solving as it breaks down complex expressions, allowing us to understand the roots of the equation more intuitively.

Quadratic Formula

The quadratic formula is a universal tool for finding the solutions of any quadratic equation in the form of \( ax^2 + bx + c = 0 \). This formula is beneficial when factoring is difficult or not possible. The quadratic formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

For the equation \( 2x^2 + 7x + 5 = 0 \), the coefficients are \( a = 2 \), \( b = 7 \), and \( c = 5 \). Plug these into the formula:

• \( x = \frac{-7 \pm \sqrt{(7)^2 - 4 \times 2 \times 5}}{2 \times 2} \)

Calculate the discriminant \( b^2 - 4ac \), where if it's positive, there are two real solutions; if zero, one real solution; and if negative, the solutions are not real. In this case, the negative discriminant reveals no real roots, aligning with the fact that the quadratic doesn't cross the x-axis. This emphasizes the role of the quadratic formula in providing a complete assessment of potential solutions.

Checking Solutions

Checking solutions is crucial to confirm the correctness of results obtained from solving quadratic equations. There are multiple methods to ensure your solutions are accurate, each allowing you to double-check your work.

One common approach is substitution. Substitute the solutions back into the original equation \((x+1)(2x+5) = -1\) and verify if both sides of the equation match. In the case of complex or incorrect solutions, the sides will not equate, indicating errors or inapplicability. Another method is graphing, which visually shows the x-intercepts. An absence of real intercepts as in our exercise indicates non-real solutions for the equation given, reaffirming the solution set determined by the quadratic formula.

Both methods underscore the importance of verification in mathematical problem-solving, preventing missteps and ensuring the accuracy of results.