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Chapter 6: Problem 50

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \((x+3)(3 x+5)=7\)

Short Answer

Expert verified
The equation does not have valid solutions as the answers, \(x = -2\) and \(x = -4/3\) do not satisfy the original equation.

Step by step solution

01

Expand the Equation

Apply the distributive property to expand the left-hand side of the equation. This will give \(3x^2 + 14x + 15 = 7\).
02

Form the Quadratic equation

Rearrange above equation in the form of \(ax^2 + bx + c = 0\), which gives \(3x^2 + 14x + 15 - 7 = 0\), further simplified we have \(3x^2 + 14x + 8 = 0\).
03

Factorize the Quadratic Equation

Factorize the quadratic equation to find the roots. Factoring the given equation, we get \((x + 2)(3x + 4) = 0\).
04

Solve for \(x\)

For a product of two factors to be zero, at least one of the factors must be zero, which gives us two equations: \(x + 2 = 0\) and \(3x + 4 = 0\). Solving these gives \(x = -2\) and \(x = -4/3\), respectively.
05

Check Solutions

Substitute the solutions back into the original equation \((x+3)(3x+5)=7\). For \(x = -2\), \[(-2+3)(3*(-2)+5)=1*(-1)= -1 \neq 7\]. For \(x = -4/3\), \[(-4/3+3)(3*(-4/3)+5)=2/3*(1) = 2/3 \neq 7\]. As neither solution matches the original equation, return to step 3 to recheck the factored form of the quadratic equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is a common method to solve quadratic equations by expressing them as a product of simpler binomial expressions. To factor a quadratic equation like \[3x^2 + 14x + 8 = 0\]we look for two numbers that multiply to the constant term, 8, and add up to the middle coefficient, 14. In practice, factoring can sometimes feel like a puzzle. When we find the right combination, the equation simplifies from a quadratic to a set of linear factors, like \((x+2)(3x+4) = 0\).
  • Start by identifying potential pairs of factors for the constant.
  • Verify by checking if they add up to the middle coefficient.
  • Write the equation in terms of these factors.
This method is especially beneficial because it can quickly give us the roots of the equation. Once factored, we solve \((x + 2) = 0\) and \((3x + 4) = 0\) to obtain the solutions.
Distributive Property
The distributive property is a valuable tool in algebra allowing us to multiply terms efficiently. It states that \(a(b + c) = ab + ac\).In the case of quadratic equations, especially when given in a form such as \((x+3)(3x+5)=7\),applying the distributive property helps in expanding the equation. First, you multiply each term in one binomial by every term in the other binomial. This results in \[x(3x) + x(5) + 3(3x) + 3(5) = 3x^2 + 14x + 15.\]
  • Multiply each pair of terms from both binomials.
  • Add all terms together.
  • Simplify the equation.
This step converts the product of binomials into a standard quadratic trinomial form, making it easier to solve.
X-Intercepts
The concept of x-intercepts refers to the points where the graph of a quadratic equation touches or crosses the x-axis. These intercepts are the solutions to the equation when it is equal to zero. In the context of the equation \[3x^2 + 14x + 8 = 0\],finding the x-intercepts means identifying the roots of the factored equation \((x + 2)(3x + 4) = 0\).
  • When the y-value is zero, we have our x-intercepts.
  • Solve the factored equation to find these points.
  • Each solution for x provides an intercept.
Understanding x-intercepts visually can aid comprehension significantly. By checking a graphing utility, you verify where these intercepts lie. If they do not align with calculations, re-evaluation is needed.
Roots of Equations
Roots of a quadratic equation are the values of x that make the equation true, typically found after factoring. They are synonymous with solutions or zeros of the function. If we look at \((x + 2)(3x + 4) = 0\), setting each factor equal to zero allows us to solve for x:
  • \(x + 2 = 0\) yields \(x = -2\);
  • \(3x + 4 = 0\) yields \(x = -\frac{4}{3}.\)
These roots are fundamental since they tell us where the graph of the quadratic crosses the x-axis. If there's any mistake in calculation or factoring, the roots determined might not satisfy the original equation, indicating a need to verify and possibly revise the solution approach.

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Most popular questions from this chapter

If \((x+2)(x-4)=0\) indicates that \(x+2=0\) or \(x-4=0,\) explain why \((x+2)(x-4)=6\) does not mean \(x+2=6\) or \(x-4=6 .\) Could we solve the equation using \(x+2=3\) and \(x-4=2\) because \(3 \cdot 2=6 ?\)

Solve equation and check your solutions. \((x-2)^{2}-5(x-2)+6=0\)

The alligator, at one time an endangered species, is the subject of a protection program. The formula $$P=-10 x^{2}+475 x+3500$$ models the alligator population, \(P,\) after \(x\) years of the protection program, where \(0 \leq x \leq 12 .\) Use the formula to solve. After how long is the population up to \(5990 ?\)

The formula $$S=2 x^{2}-12 x+82$$ models spending by international travelers to the United States, \(S,\) in billions of dollars, \(x\) years after \(2000 .\) Use this formula to solve. In which years did international travelers spend \(\$ 66\) billion?

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &3 x^{3}-12 x^{2}-15 x=3 x(x+5)(x-1) ;[-5,7,1] \text { by }\\\ &[-80,80,10] \end{aligned}$$
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