Completing the Square
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The document explains how to solve quadratic equations by completing the square. It defines a perfect square trinomial as having the form x^2 + bx + c, where c is the square of half of b. It provides steps for completing the square, which involves adding a constant term to both sides of the equation such that the left side becomes a perfect square trinomial that can be factorized. This process results in the solution(s) to the quadratic equation. Two examples demonstrating this process are included.
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Quadratic equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
Completing the square if a
The document provides 3 examples of solving quadratic equations by setting them equal to zero and using the quadratic formula. Each example shows the step-by-step work of isolating the constant term, factoring the equation, taking the square root of both sides to solve for the roots, and checking the solutions. The examples demonstrate how to solve quadratic equations from setting them equal to zero through finding the solution set.
15.3---Graphs-of-Quad-Functions.pptx
The document provides information about quadratic functions including:
- How to find the roots, intercept, turning point, and whether the turning point is a maximum or minimum value by analyzing the quadratic equation.
- Examples of using factorizing, completing the square, and setting the equation equal to 0 to find the roots.
- How to sketch the graph of a quadratic function using its equation by plotting the roots, intercept, turning point, and labeling whether the turning point is a max or min.
Factoring Perfect Square Trinomial - SIM
This document contains information and examples about perfect square trinomials in algebra. It defines a perfect square trinomial as having a first term and last term that are perfect squares, with a middle term that is twice the product of the square roots of the first and last terms. Examples are provided to illustrate how to identify if an expression is a perfect square trinomial or not. Several activity cards are included that provide practice identifying, factoring, and giving the factors of perfect square trinomial expressions.
perfect square trinomial
This document provides instruction on perfect square trinomials including defining them, identifying them, factoring them, and working practice problems. It begins by defining a perfect square trinomial as the result of squaring a binomial with the first and last terms being perfect squares and the middle term being twice the product of the square roots of the first and last terms. Examples are provided to illustrate. The document then provides guidance, activities, and an assessment to practice identifying, factoring, and working with perfect square trinomials.
March 7
1. The document discusses factoring perfect square trinomials, which are polynomials where the first term is a perfect square, the third term is a perfect square, and the coefficient of the second term is twice the square root of the product of the first and third term coefficients.
2. To factor a perfect square trinomial, take the square root of the first term, add it to the square root of the third term, and place it in parentheses twice to get the two factors.
3. Examples are provided of determining if a trinomial is a perfect square and factoring perfect square trinomials like x2 + 10x + 25 as (x + 5)2.
Math1
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Solving quadratic equations[1]
The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations arising from applications like finding the dimensions of a rectangle given its area. Sample problems are worked through step-by-step to find the solutions or roots of quadratic equations.
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Completing the Square
- 1. Solving Quadratic Equations By Completing the Square
- 2. What is a Perfect Square Trinomial?
- 4. Perfect Square Trinomials 100 4 25/4
- 5. How to “Complete the Square” Solve the following equation by completing the square: x 2 + 8x – 20 = 0 Step 1: Move quadratic term, and linear term to left side of the equation x 2 + 8x = 20
- 6. Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. x 2 + 8x + ? = 20 + ? x 2 + 8x + 16 = 20 + 16 8/2 = 4 4 2 = 16
- 7. Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
- 8. Step 4: Take the square root of each side and solve. x = 10 or x = -2
- 9. Example #2 2 2 2
- 10. Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. x 2 – 7/2 + 49/16 = -6 + 49/16 x 2 – 7/2 + 49/16 = -96/16 + 49/16 x 2 – 7/2 + 49/16 = -47/16 ½ * 7/2 = 7/4 (7/4) 2 = 49/16