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Chapter 10: Problem 46

Solve the equation. Tell which solution method you used. \(34 x^{4}-85 x^{3}+51 x^{2}=0\)

Short Answer

Expert verified
The solutions for the equation \(34 x^{4}-85 x^{3}+51 x^{2}=0\) are \(x = 0, x = \frac{51}{34}, x = 1 \). The solution method used is factoring.

Step by step solution

01

Find out the Greatest common factor

The given equation is \(34 x^{4}-85 x^{3}+51 x^{2}=0\). First, find the greatest common factor of all the terms in the given equation. Here, \(x^{2}\) is the greatest common factor. Factor out \(x^{2}\) and rewrite the equation as: \(x^{2}(34x^{2} - 85x + 51) = 0\).
02

Factor the Quadratic Equation

Next, factor the quadratic equation \((34x^{2} - 85x + 51)\). We need to find two numbers whose product is \(34*51 = 1734\) and whose sum is \(-85\). After calculation we find that \(-34\) and \(-51\) are those two numbers. \(34x^{2} - 85x + 51 = 34x^{2} - 34x - 51x + 51 = 34x(x - 1) - 51(x - 1) = (34x - 51)(x - 1)\). So now we have \(x^{2}*(34x - 51)*(x - 1) = 0\).
03

Solve Each Equation for x

Now solve each of the factored equations for x. Set each factor equal to zero. Here we got three equations: \(x^{2} = 0, 34x - 51 = 0, x - 1 = 0\). Solving these equations will give us x = 0, x = \frac{51}{34}, x = 1.

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Key Concepts

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

Factoring
Factoring is a key technique used to solve polynomial equations. It involves breaking down a complex expression into simpler products of factors. By doing so, we can more easily solve equations and identify the roots of polynomials.
  • In factoring, we look for common terms or values that can be extracted from each part of the equation, making it cleaner and simpler.
  • For example, in the equation given, we first extracted the greatest common factor, which was \(x^2\).
  • This allowed us to express \(34x^4 - 85x^3 + 51x^2\) as \(x^2(34x^2 - 85x + 51)\).
By factoring, we transform the equation into a form that's easier to solve. Remember, once an equation is factored fully, setting each factor to zero helps find the equation's solutions easily.
Quadratic Equations
Quadratic equations are polynomials of degree two, typically expressed in the form \(ax^2 + bx + c = 0\). Solving quadratic equations can often involve factoring, using the quadratic formula, or completing the square.
  • In our context, after factoring out the greatest common factor, what remained was a quadratic expression: \(34x^2 - 85x + 51\).
  • To factor this quadratic, we identify two numbers which multiply to give the product of the leading coefficient and the constant (\(34 \times 51 = 1734\)), and add up to the middle coefficient (\(-85\)).
  • These numbers were \(-34\) and \(-51\), allowing us to rewrite and factor the quadratic as \((34x - 51)(x - 1)\).
Understanding how to manipulate and solve quadratic equations is essential in algebra, paving the way to tackle more complex problems.
Greatest Common Factor
The greatest common factor (GCF) is the largest factor shared by all terms in an expression or equation. Finding the GCF simplifies expressions and makes equations easier to solve.
  • To find the GCF, identify common components in each term of the equation. For \(34x^4 - 85x^3 + 51x^2\), each term shared \(x^2\) as a factor.
  • By factoring out \(x^2\), we simplified the original equation to \(x^2(34x^2 - 85x + 51) = 0\).
  • This simplification reduces the degree of the polynomial, allowing us to focus on the more manageable quadratic expression that remains.
Always look for the GCF first when attempting to solve polynomial equations, as it can significantly simplify the problem-solving process.

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