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Magazine Chapter 6: Problem 29
Solve. See Examples 3 through 6. $$ (x+4)(x-9)=4 x $$
Short Answer
Expert verified
The solutions are \(x = 12\) and \(x = -3\).
Step by step solution
01
Distribute the Left Side
Expand the expression \((x + 4)(x - 9)\) using the distributive property (FOIL method). This gives:\[x(x - 9) + 4(x - 9) = x^2 - 9x + 4x - 36\] Combine like terms to simplify:\[x^2 - 5x - 36\]
02
Set the Equation to Zero
Now take the simplified form of the left side, \(x^2 - 5x - 36\), and set it equal to the left side of the equation:\[x^2 - 5x - 36 = 4x\]To solve the quadratic equation, move all terms to one side to set the equation to zero:\[x^2 - 5x - 36 - 4x = 0\]Simplify to:\[x^2 - 9x - 36 = 0\]
03
Solve the Quadratic Equation by Factoring
Try to factor the quadratic equation \(x^2 - 9x - 36 = 0\). Look for two numbers whose product is -36 and sum is -9. These numbers are -12 and 3.Write the factored form:\[(x - 12)(x + 3) = 0\]
04
Find the Solutions
Set each factor equal to zero to find the solutions for \(x\):1. \(x - 12 = 0\) Solve for \(x\): \[x = 12\]2. \(x + 3 = 0\) Solve for \(x\): \[x = -3\]
05
Verify the Solutions
Check each solution by substituting back into the original equation \((x + 4)(x - 9) = 4x\). For \(x = 12\):- Left side: \((12 + 4)(12 - 9) = 16 \times 3 = 48\)- Right side: \(4 \times 12 = 48\)For \(x = -3\):- Left side: \((-3 + 4)(-3 - 9) = 1 \times (-12) = -12\)- Right side: \(4 \times (-3) = -12\)Both solutions satisfy the original equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Property
The distributive property is a key mathematical principle used to expand expressions. In this problem, we begin by expanding \((x + 4)(x - 9)\) using the distributive property, often remembered as the FOIL method.
To apply it, follow these steps:
Finally, simplify by combining like terms. In this case, \(-9x + 4x = -5x\). So, the expression expands to:\[x^2 - 5x - 36\]
To apply it, follow these steps:
- First, multiply the first terms: \(x \times x = x^2\)
- Outside terms: \(x \times (-9) = -9x\)
- Inside terms: \(4 \times x = 4x\)
- Last terms: \(4 \times (-9) = -36\)
Finally, simplify by combining like terms. In this case, \(-9x + 4x = -5x\). So, the expression expands to:\[x^2 - 5x - 36\]
Factoring Quadratics
Factoring quadratics is a method used to find the roots or solutions of quadratic equations. After simplifying our equation to the form \(x^2 - 9x - 36 = 0\), we need to factor it.
We search for two numbers which multiply to \(-36\) and add to \(-9\). This kind of problem-solving often requires some trial and error or strategic guesswork.
We search for two numbers which multiply to \(-36\) and add to \(-9\). This kind of problem-solving often requires some trial and error or strategic guesswork.
- In this case, the numbers are \(-12\) and \(3\).
- These numbers satisfy the conditions: \(-12 \times 3 = -36\) and \(-12 + 3 = -9\).
FOIL Method
The FOIL method is an acronym that stands for First, Outside, Inside, Last. It's a specific application of the distributive property for multiplying two binomials.
To understand FOIL method, let's apply it to our binomials:
To understand FOIL method, let's apply it to our binomials:
- First: Multiply the first terms: \(x \times x = x^2\)
- Outside: Multiply the outer terms: \(x \times (-9) = -9x\)
- Inside: Multiply the inner terms: \(4 \times x = 4x\)
- Last: Multiply the last terms: \(4 \times (-9) = -36\)
Solution Verification
Verification is the final step in solving any equation. After finding possible solutions \(x = 12\) and \(x = -3\), it鈥檚 important to substitute them back into the original equation to ensure their validity.
For \(x = 12\):
For \(x = -3\):
This step ensures that our solutions are consistent with the original equation, confirming their correctness.
For \(x = 12\):
- The left side: \((12 + 4)(12 - 9) = 16 \times 3 = 48\)
- The right side: \(4 \times 12 = 48\)
For \(x = -3\):
- The left side: \((-3 + 4)(-3 - 9) = 1 \times (-12) = -12\)
- The right side: \(4 \times (-3) = -12\)
This step ensures that our solutions are consistent with the original equation, confirming their correctness.
