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Chapter 0: Problem 103

Solve the quadratic equation by the method of your choice. $$3 x^{2}-12 x+12=0$$

Short Answer

Expert verified
The solutions to the quadratic equation are \(x = 2\) and \(x = 2\).

Step by step solution

01

Identify the coefficients

Here, in equation \(3x^2 - 12x + 12 = 0\) the coefficient of \(x^2\) is 'a' which is 3, the coefficient of 'x' is 'b' which is -12, and the constant term is 'c' which is 12.
02

Plug the values into the Quadratic Formula

The quadratic formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values of a, b and c into the formula. That leads to \(x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4*3*12}}{2*3}\).
03

Simplify within the square root

Firstly simplify the terms in the square root: \(\sqrt{(-12)^2 - 4*3*12} = \sqrt{144 - 144} = \sqrt{0}\).
04

Simplify further

Now, substitute \(\sqrt{0}\) back into the quadratic formula: \(x = \frac{12 \pm 0}{6}\). Now, we have two solutions, \(x = \frac{12 + 0}{6}\) and \(x = \frac{12 - 0}{6}\). Both of these simplify down to 2.

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Key Concepts

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

Quadratic Formula
The Quadratic Formula is a proven and reliable method for finding the roots of any quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \). This formula is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It incorporates the coefficients of the quadratic equation, represented by \( a \) for the quadratic term, \( b \) for the linear term, and \( c \) for the constant term, to ultimately solve for the variable \( x \).

Coefficients Identification
Understanding how to accurately identify coefficients is crucial in solving quadratic equations. In the given equation \( 3x^2 - 12x + 12 = 0 \), the coefficients are the numerical components before the variable terms: \( a = 3 \) is in front of \( x^2 \), \( b = -12 \) is in front of \( x \), and \( c = 12 \) is the constant term without any variables.

Identifying Coefficients

  • Quadratic term coefficient (a): 3
  • Linear term coefficient (b): -12
  • Constant term (c): 12

Equation Simplification
Simplifying an equation makes it easier to solve. In our exercise, simplification involves several steps, including squaring the coefficient \( b \) and multiplying \( 4ac \), as shown in the quadratic formula application. Specifically, within the square root, the simplification of \( b^2 - 4ac \) is important since it determines the nature of the roots鈥攔eal or complex. In this case, the simplification results in the square root of zero, which suggests that our equation has a real and repeated root. The equation simplifies to \( x = \frac{12 \pm 0}{6} \), which further simplifies to a single solution: \( x = 2 \).

Polynomial Roots
The roots or solutions of a polynomial equation are the values of the variable that satisfy the equation. In other words, these are the points at which the graph of the equation crosses the x-axis. In the context of quadratic equations, there can be either two distinct real roots, one real repeated root, or two complex roots. The nature of the roots is inferred from the discriminant, \( b^2 - 4ac \). For the equation \( 3x^2 - 12x + 12 = 0 \), there is one real root, \( x = 2 \), repeated twice. This is because the discriminant is zero, indicating that the graph of the equation touches the x-axis at just one point鈥攃orresponding to the repeated root.

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Most popular questions from this chapter

What is a perfect square trinomial and how is it factored?

Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial.

What is the discriminant and what information does it provide about a quadratic equation?

Describe ways in which solving a linear inequality is similar to solving a linear equation.

Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by \(30 \%\) Now the sale price is reduced by another \(30 \% .\) If \(x\) is the television's original price, the sale price can be modeled by $$(x-0.3 x)-0.3(x-0.3 x)$$ a. Factor out \((x-0.3 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(30 \%\) reduction followed by a \(30 \%\) reduction, is the television selling at \(40 \%\) of its original price? If not, at what percentage of the original price is it selling?
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