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Chapter 10: Problem 97

Solve each equation. $$ x(x+3)=40 $$

Short Answer

Expert verified
x = -8, 5

Step by step solution

01

- Expand the Equation

Expand the equation by applying the distributive property: \[ x(x + 3) = 40 \ \Rightarrow x^2 + 3x = 40 \]
02

- Bring Equation to Standard Form

Rewrite the equation in standard quadratic form: \[ x^2 + 3x - 40 = 0 \]
03

- Factor the Quadratic Equation

Factor the quadratic equation. Find two numbers that multiply to -40 and add up to 3: \[ (x + 8)(x - 5) = 0 \]
04

- Solve for x

Set each factor equal to zero and solve for x: \[ x + 8 = 0 \ x = -8 \] \[ x - 5 = 0 \ x = 5 \]

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Key Concepts

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

Distributive Property
To solve the given quadratic equation, we first need to expand it using the distributive property. The distributive property states that for any numbers a, b, and c: \[ a(b + c) = ab + ac \] Applying this to our problem: \[ x(x + 3) = 40 \] Distribute the x across the terms inside the parentheses: \[ x \times x + x \times 3 = 40 \] Simplifying this: \[ x^2 + 3x = 40 \] This new equation allows us to move on to the next steps to solve the quadratic equation.
Standard Form
The next step is to bring the expanded equation to the standard form of a quadratic equation. A quadratic equation in standard form looks like: \[ ax^2 + bx + c = 0 \] Comparing this to our equation, we have: \[ x^2 + 3x = 40 \] To get all terms on one side, subtract 40 from both sides of the equation: \[ x^2 + 3x - 40 = 0 \] Now our equation is in standard form, where: \[ a = 1, \ b = 3, \ c = -40 \] With this form, we can now move on to the next step, which involves factoring.
Factoring Quadratics
In this step, we need to factor the quadratic equation. We look for two numbers that multiply to the constant term, -40, and add up to the coefficient of the linear term, 3. Here is how to approach this:
  • Find numbers that multiply to -40.
  • Check which pair of those numbers adds up to 3.
The correct pairs are 8 and -5: \[ 8 \times (-5) = -40 \] and \[ 8 + (-5) = 3 \] Using these numbers, we rewrite the quadratic equation as a product of two binomials: \[ (x + 8)(x - 5) = 0 \] This is the factored form of the quadratic equation.
Solving for x
The final step is to solve for x by setting each factor equal to zero. We have: \[ (x + 8)(x - 5) = 0 \] Set each factor to zero: \[ x + 8 = 0 \] Solve for x: \[ x = -8 \] and: \[ x - 5 = 0 \] Solve for x: \[ x = 5 \] Therefore, the solutions to the equation are: \[ x = -8 \] and \[ x = 5 \] These are the values of x that satisfy the original equation \[ x(x+3)=40 \].

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Most popular questions from this chapter

Simplify by first converting to rational exponents. Assume that all variables represent positive real numbers. See Example 4 $$ 3^{1 / 2} \cdot 3^{3 / 2} $$

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