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Chapter 4: Problem 78

Solve the quadratic equation by factoring. $$ 2 x^2-10 x-72=0 $$

Short Answer

Expert verified
The solutions are \(x = 9\) and \(x = -4\).

Step by step solution

01

Simplify The Equation

The equation needs to be made simpler if possible. Given \(2x^2 - 10x - 72 = 0\), we find that all the terms of the equation are divisible by 2, so let's simplify: \((2x^2 - 10x - 72) / 2 = 0 / 2\), which gives us \(x^2 - 5x - 36 = 0\) as the simplified equation.
02

Factorize The Quadratic Equation

This step requires finding two numbers that add to -5 (the coefficient of x) and multiply to -36 (the constant term). Such numbers are -9 and +4. So, the factorised form of the equation becomes \((x - 9)(x + 4) = 0\).
03

Solve Each Resulting Equation

This step involves equating each factor to zero and solving for x. (1) \(x - 9 = 0 \Rightarrow x = 9\), (2) \(x + 4 = 0 \Rightarrow x = -4\). So, the solutions to the quadratic equation are x = 9 and x = -4.

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Key Concepts

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

Factoring Quadratics
Factoring quadratics involves expressing the quadratic equation as a product of two binomial expressions. In our example, after simplifying the equation to \(x^2 - 5x - 36 = 0\), we need to identify two numbers that multiply to the constant term \(-36\) and also add up to the coefficient of the linear term \(-5\).

Steps to factor:
  • Identify the constant term (here, \(-36\)).
  • Identify the linear coefficient (here, \(-5\)).
  • Find two numbers that multiply to \(-36\) and add to \(-5\). Those numbers are \(-9\) and \(+4\).
  • Rewrite the quadratic as a product of linear factors \((x - 9)(x + 4)\).
Factoring is a useful technique because it often transforms the equation into a form that's easier to solve. By writing the equation as \((x - 9)(x + 4) = 0\), we now have a straight path to finding the roots.
Simplifying Equations
Simplifying equations is about reducing the equation to its simplest, most manageable form. Initially, we had the equation \(2x^2 - 10x - 72 = 0\). By examining the equation, we discovered a common factor of 2 among all terms. This simplifies our calculations greatly.

Here鈥檚 how it's done:
  • Divide each term of the equation by its greatest common factor (in this case, 2).
  • Rewrite the equation with simplified coefficients: \(x^2 - 5x - 36 = 0\).

Simplifying the equation is crucial, especially for larger numbers, as it makes the process of solving far more straightforward. It prevents errors and speeds up calculations.
Roots of Quadratic Equations
Finding the roots of a quadratic equation means determining the values of \(x\) for which the equation equals zero. After factoring the equation into \((x - 9)(x + 4) = 0\), we have two factors that can each be set to zero to solve for \(x\).

Solving for roots:
  • Take each factor separately: \(x - 9\) and \(x + 4\).
  • Solve each equation for \(x\): \(x - 9 = 0\) leads to \(x = 9\), and \(x + 4 = 0\) leads to \(x = -4\).
The roots \(x = 9\) and \(x = -4\) are the solutions to the original quadratic equation, representing the values for \(x\) that satisfy \(2x^2 - 10x - 72 = 0\). Understanding how to find these roots provides insight into the behavior of quadratic functions and their graphs.

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Most popular questions from this chapter

Determine whether the binomial is a factor of the polynomial function. $$ h(x)=6 x^5-15 x^4-9 x^3 ; x+3 $$

Use the method of your choice to factor the polynomial completely. Explain your reasoning. $$ 16 n^4-1 $$

Factor the polynomial completely. $$ y^4-3 y^2-28 $$

\(f(x)=(2 x-3)^3 ; g(x)=8 x^3-36 x^2+54 x-27\)

\(h(x)=(x+2)^5\), $$ k(x)=x^5+10 x^4+40 x^3+80 x^2+64 x $$
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