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Chapter 13: Problem 42

Review solving equations. Solve $$ x^{2}+4 x=60[6.2] $$

Short Answer

Expert verified
The solutions are x = -10 and x = 6.

Step by step solution

01

Set the Equation to Zero

First, move all terms to one side of the equation to set it to zero. Subtract 60 from both sides: x^2 + 4x - 60 = 0
02

Factor the Quadratic Equation

Next, factor the quadratic equation. Find two numbers that multiply to -60 and add up to 4. These numbers are 10 and -6, hence the equation can be factored as: (x + 10)(x - 6) = 0
03

Set Each Factor to Zero and Solve

Now, set each factor equal to zero and solve for x: For x + 10 = 0x = -10. For x - 6 = 0x = 6.
04

Verify the Solutions

Finally, verify the solutions by plugging them back into the original equation:For x = -10, (-10)^2 + 4(-10)= 100 - 40 = 60which is true.For x = 6, (6)^2 + 4(6) = 36 + 24 = 60which is also true.

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Key Concepts

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

setting equations to zero
To solve a quadratic equation, the first step is to set it to zero. This involves moving all terms to one side of the equation.
Doing this ensures we can use factoring or the quadratic formula later.
For our exercise \(x^{2}+4x=60\), we begin by subtracting 60 from both sides to get:
\[ x^2 + 4x - 60 = 0 \]
This step is crucial because we need a zero on one side to factor the equation correctly.
factoring quadratic equations
Once the equation is set to zero, the next step is factoring the quadratic equation.
Factoring involves finding two numbers that multiply to the constant term (-60 in this example) and add up to the coefficient of the middle term (4 in this example).
For \[ x^2 + 4x - 60 = 0 \], we try to find two numbers that multiply to -60 and add up to 4.
These numbers are 10 and -6. So, we can write the equation as two binomials:
\[ (x + 10)(x - 6) = 0 \],
This factored form represents the equation of the quadratic function as two expressions.
verifying solutions
After factoring and solving for the variables, the final step is verifying the solutions.
This ensures that the solutions satisfy the original equation.
For our factored equation \[ (x + 10)(x - 6) = 0 \], solving for x gives us two solutions: \(x = -10\) and \(x = 6\).
To verify, substitute these values back into the original equation:
  • For \(x = -10\): \[ (-10)^2 + 4(-10) = 100 - 40 = 60 \]
  • For \(x = 6\): \[ (6)^2 + 4(6) = 36 + 24 = 60 \]
Both solutions satisfy the original equation, thus confirming their correctness.

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