Problem 57 Solve each equation. $$x^{2}+5... [FREE SOLUTION]

Chapter 5: Problem 57

Solve each equation. $$x^{2}+5 x=6$$

Short Answer

Expert verified

The solutions are $ x = 1 $ and $ x = -6 $.

Step by step solution

Identify the quadratic equation

Recognize the given equation as a quadratic equation of the form $ ax^2 + bx + c = 0 $. The initial equation is $ x^2 + 5x - 6 = 0 $.

Factor the quadratic equation

Find two numbers that multiply to -6 (the constant term $c$) and add to 5 (the coefficient of $x$'s term). The numbers -1 and 6 work because $ -1 \times 6 = -6 $ and $ -1 + 6 = 5 $. Thus, factor the equation: $ (x - 1)(x + 6) = 0 $.

Apply the zero-product property

Use the zero-product property which states that if $ (x - 1)(x + 6) = 0 $, then one or both of the factors must be zero. This gives the equations: $ x - 1 = 0 $ and $ x + 6 = 0 $.

Solve for $x$

Solve each equation separately: For $ x - 1 = 0 $, add 1 to both sides to get $ x = 1 $. For $ x + 6 = 0 $, subtract 6 from both sides to get $ x = -6 $. Therefore, the solutions are $ x = 1 $ and $ x = -6 $.

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Key Concepts

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

Factoring Quadratic Equations

Factoring quadratic equations is a crucial step in solving them. A quadratic equation typically looks like this: \[ ax^2 + bx + c = 0 \].
In our example, we start with: \[ x^2 + 5x - 6 = 0 \]. This is already in the standard quadratic form.
To factor this, we need to find two numbers that multiply together to give us the constant term $c = -6$ and add up to the coefficient of the $x$ term $b = 5$.
In our case, the numbers are -1 and 6. To confirm, $ -1 \times 6 = -6$, and $ -1 + 6 = 5$. Now we can write $ x^2 + 5x - 6 = (x - 1)(x + 6)$ as the factored form.
Factoring is breaking down the equation into simpler expressions, which will help in the next steps.

Zero-Product Property

Once we have factored the quadratic equation, we apply the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero.
For the factored form $ (x - 1)(x + 6) = 0$, we set each factor to zero separately:

\[ x - 1 = 0 \]
\[ x + 6 = 0 \]

The idea here is simple. If two things multiply to zero, one or both of them have to be zero. So, we turn our attention to solving these smaller, simpler equations.

Solving for Roots

The final step is to solve for the roots of each equation resulting from the zero-product property.
We have two small equations to solve:

From $ x - 1 = 0$, we add 1 to both sides, resulting in $ x = 1$.
From $ x + 6 = 0$, we subtract 6 from both sides, resulting in $ x = -6$.

This means the roots, or solutions, of the original quadratic equation $ x^2 + 5x - 6 = 0$ are $ x = 1 $ and $ x = -6$.
The roots are the values of x that make the original equation true.

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91影视

Solve each equation. $$x^{2}+5 x=6$$

Short Answer

Step by step solution

Identify the quadratic equation

Factor the quadratic equation

Apply the zero-product property

Solve for \(x\)

Key Concepts

Factoring Quadratic Equations

Zero-Product Property

Solving for Roots

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