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Chapter 1: Problem 37

Solve the equation by factoring. $$x^{2}+x-12=0$$

Short Answer

Expert verified
The solutions are \(x = -4\) and \(x = 3\).

Step by step solution

01

Write Down Quadratic Equation

The given equation is \(x^2 + x - 12 = 0\). We need to solve this by factoring.
02

Identify Two Numbers

Find two numbers whose product is \(-12\) (the constant term) and whose sum is \(1\) (the coefficient of \(x\)). These numbers are \(4\) and \(-3\) because \(4\times(-3) = -12\) and \(4 + (-3) = 1\).
03

Write the Expression in Factored Form

Use the numbers found to rewrite the quadratic expression as \((x + 4)(x - 3)\). Thus, the equation becomes \((x + 4)(x - 3) = 0\).
04

Solve Each Factor for Zero

Set each factor equal to zero. So, solve the equations \(x + 4 = 0\) and \(x - 3 = 0\).
05

Solve for x

For \(x + 4 = 0\), subtract \(4\) from both sides to get \(x = -4\). For \(x - 3 = 0\), add \(3\) to both sides to get \(x = 3\).

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Key Concepts

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

Factoring Method
The factoring method is a powerful technique for solving quadratic equations. It's like solving a puzzle where you break down the expression into simpler pieces. In a standard quadratic equation, of the form \( ax^2 + bx + c = 0 \), here's how the method works:
  • First, we look for two numbers that multiply together to give the constant term, \( c \), and at the same time add up to the coefficient of the \( x \) term, \( b \).
  • In our exercise, the equation is \( x^2 + x - 12 = 0 \).
  • Here, we need two numbers that multiply to \(-12\) (constant term) and add to \(1\) (coefficient of \( x \)).
  • These numbers turn out to be \( 4 \) and \(-3 \).
  • Once found, these numbers allow us to express the quadratic in its factored form, which looks like \((x + a)(x - b) = 0\).
Using this method effectively reduces a quadratic equation into a form where its solutions become more apparent.
Roots of Equations
The roots of a given equation are the values of \( x \) that make the equation true. For quadratic equations, these roots are also known as solutions or zeros.
  • When the equation is factored, as in our example \((x + 4)(x - 3) = 0\), determining the roots becomes more straightforward.
  • The principle we use is that if a product of two terms equals zero, then at least one of these terms must be zero.
  • Thus, we set each factor of the product equal to zero separately, leading to \( x + 4 = 0 \) and \( x - 3 = 0 \) in our case.
  • Solving these simple linear equations gives us the roots of the quadratic equation, which in this example are \( x = -4 \) and \( x = 3 \).
Finding the roots is crucial as it represents the values where the original quadratic equation graphically intersects the x-axis.
Solving Quadratic Equations
Solving quadratic equations can be likened to uncovering the secret numbers that satisfy the equation for given values of \( x \). The steps are consistent but require attention to detail.
  • Identify the quadratic equation you need to solve. It generally takes the form \( ax^2 + bx + c = 0 \).
  • Choose an appropriate method: factoring, using the quadratic formula, or completing the square are common ways.
  • In our exercise, we used the factoring method to simplify \( x^2 + x - 12 = 0 \) into \((x + 4)(x - 3) = 0\).
  • This approach transforms the problem into solving simpler equations that lead directly to the solutions.
  • After factoring, tackle each term independently to find the values of \( x \).
Remember, experimentation with different methods is often necessary based on the equation's structure. Additionally, each solved equation gives us a deeper understanding of the relationships in algebra.

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Most popular questions from this chapter

A trinomial of the form \(x^{4}+a x^{2}+b\) can sometimes be factored easily. For example, \(x^{4}+3 x^{2}-4=\left(x^{2}+4\right)\left(x^{2}-1\right) .\) But \(x^{4}+3 x^{2}+4\) cannot be factored in this way. Instead, we can use the following method. $$\begin{aligned}x^{4}+3 x^{2}+4 &=\left(x^{4}+4 x^{2}+4\right)-x^{2} \\\&=\left(x^{2}+2\right)^{2}-x^{2} \\\&=\left[\left(x^{2}+2\right)-x\right]\left[\left(x^{2}+2\right)+x\right] \\\&=\left(x^{2}-x+2\right)\left(x^{2}+x+2\right)\end{aligned}$$

Perform the indicated operations and simplify. $$(1-b)^{2}(1+b)^{2}$$

In the expression \(2 / \sqrt{x}\) we would eliminate the radical if we were to square both numerator and denominator. Is this the same thing as rationalizing the denominator?

Factor the expression completely. $$x^{2}-36$$

Sketch the region given by the set. $$\left\\{(x, y) | x^{2}+y^{2}>4\right\\}$$
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