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Magazine Chapter 10: Problem 33
Solve the quadratic equation by the most convenient method. $$x^{2}-20 x+100=0$$
Short Answer
Expert verified
The solution to the quadratic equation is \( x = 10 \).
Step by step solution
01
Identify a, b, and c
From the quadratic equation \( x^2 - 20x + 100 = 0 \), identify the coefficients as: \( a = 1 \), \( b = -20 \), and \( c = 100 \).
02
Check if the quadratic is a perfect square trinomial
Examine if the quadratic fits the structure of a perfect square trinomial. For a quadratic to be a perfect square trinomial, it must be in the form \( (ax)^2 - 2abx + b^2 \). Upon observing, \( (x)^2 - 2(10)x + (10)^2 \), it could be seen that it is indeed a perfect square trinomial.
03
Factor the perfect square trinomial
Factor the equation following the structure of \( (ax - b)^2 \). In this case, the factored form would be \( (x - 10)^2 \). This implies that \( x - 10 = 0 \).
04
Solve for x
Solving for \( x \) in the equation \( x - 10 = 0 \), we arrive at \( x = 10 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Trinomials
Factoring trinomials is a key skill for solving quadratic equations efficiently. When you have a quadratic equation in the form of \( ax^2 + bx + c = 0 \), one handy method is to try to factor it into the product of two binomials, like \( (px + q)(rx + s) = 0 \).
The goal of factoring is to express the quadratic in a simpler multiplied form that equals zero. This makes it easier to solve for the values of \( x \) by setting each binomial equal to zero.
The goal of factoring is to express the quadratic in a simpler multiplied form that equals zero. This makes it easier to solve for the values of \( x \) by setting each binomial equal to zero.
- First, identify the coefficients: these are the numbers in front of \( x^2 \), the linear \( x \) term, and the constant.
- Next, look for two numbers that multiply to \( ac \) (the product of the coefficient of \( x^2 \) and the constant term \( c \)), and add up to \( b \) (the coefficient of \( x \)).
- Once found, these numbers help break down the middle term into a form where factoring is possible.
Perfect Square Trinomials
A perfect square trinomial is a special type of trinomial that results from squaring a binomial.
In other words, if a trinomial can be expressed as \((ax + b)^2 = ax^2 + 2abx + b^2\), then it is a perfect square trinomial. A clear sign of this is when the first and last terms are perfect squares.
In other words, if a trinomial can be expressed as \((ax + b)^2 = ax^2 + 2abx + b^2\), then it is a perfect square trinomial. A clear sign of this is when the first and last terms are perfect squares.
- Identify the leading term and constant to see if they form the square of a binomial.
- The middle term should equal twice the product of the square roots of the first and last terms.
Solving Quadratic Equations
Solving quadratic equations means finding the value(s) of \( x \) that satisfy the equation. Quadratics are in the form \( ax^2 + bx + c = 0 \), and there are multiple methods to solve them, including factoring, using the quadratic formula, completing the square, and graphing.
- Factoring: Useful when the quadratic can be decomposed into binomials. This was our method as the equation \( (x - 10)^2 = 0 \) showcased.
- Quadratic Formula: A general method that solves any quadratic equation: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
- Completing the Square: This involves manipulating the equation into a perfect square trinomial, making it easier to solve.
- Graphing: Visual method where the solution is found at the points where the parabola crosses the x-axis.
Coefficients in Quadratic Equations
In a quadratic equation \( ax^2 + bx + c = 0 \), the terms \( a \), \( b \), and \( c \) are the coefficients, and they play important roles in determining the parabola's shape and position.
- \( a \) is the coefficient of \( x^2 \) and dictates the openness and direction of the parabola. If \( a \) is positive, the parabola opens upwards; if negative, downwards.
- \( b \) affects the parabola's symmetry and the location of its vertex along the x-axis.
- \( c \) is the constant term, impacting the vertical shift of the parabola, effectively defining where the parabola intersects the y-axis.
