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Chapter 6: Problem 55

Solve. $$ x^{2}+4 x=45 $$

Short Answer

Expert verified
The solutions are x = -9 and x = 5.

Step by step solution

01

Move the constant term

To start solving the equation Move the constant term 45 to the left side to set the equation to 0. You need to subtract 45 from both sides. {x^{2} + 4x - 45 = 0}.
02

Factor the quadratic equation

Next, factorize the quadratic equation x^{2} + 4 x - 45 into two binomials. Look for two numbers that multiply to -45 and add to 4. Those numbers are 9 and -5. The factored form is (x + 9)(x - 5) = 0.
03

Solve for x

Set each binomial equal to 0 and solve for x. Solve {(x + 9) = 0} for x to get {x = -9}. Solve {(x - 5) = 0} for x to get {x = 5}.

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Key Concepts

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

Factoring Quadratic Equations
Quadratic equations are equations of the form \( ax^{2} + bx + c = 0 \). Factoring is one way to solve them. When we factor, we break down the quadratic equation into simpler terms, usually binomials.
For our example, \( x^{2} + 4x - 45 \), we look for two numbers that multiply to -45 (the constant term) and add to 4 (the coefficient of the x term). Those numbers are 9 and -5.
This gives us the factored form: \( (x + 9)(x - 5) \). Factoring makes it easier to solve the equation by breaking it into two simpler equations.
Setting Equation to Zero
Before we can factor a quadratic equation, we need to set it to zero. This means getting all terms on one side of the equation so that the other side is zero.
In our problem, \( x^{2} + 4x = 45 \), we move the constant term 45 to the left side by subtracting 45 from both sides of the equation. This gives us: \( x^{2} + 4x - 45 = 0 \).
Setting the equation to zero is crucial because it allows us to apply the zero-product property after factoring. In any equation, if the product of two terms is zero, at least one of those terms must be zero.
Solving for x
Once the quadratic equation is factored and set to zero, solving for x becomes straightforward.
Using the factored form \( (x + 9)(x - 5) = 0 \), we apply the zero-product property. This tells us that if \( a \cdot b = 0 \), then either \( a = 0 \) or \( b = 0 \).
So, we set each binomial equal to zero: \( x + 9 = 0 \) and \( x - 5 = 0 \). Solving these, we get \( x = -9 \) and \( x = 5 \).
These are the solutions to the quadratic equation, meaning \( x = -9 \) and \( x = 5 \). Always check both solutions to ensure they satisfy the original equation.

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