Factorising quadratics, sketching and inequalities

Keywords

[table][br][tr][br][td]English[/td][br][td]Japanese[/td][br][td]Korean[/td][br][td]Chinese Simplified[/td][br][/tr][br][tr][br][td]Factorise/Factorize[/td][br][td]因数分解[/td][br][td]인수분해[/td][br][td]因式分解[/td][br][/tr][br][tr][br][td]Quadratic Equation[/td][br][td]二次方程式[/td][br][td]이차방정식[/td][br][td]二次方程[/td][br][/tr][br][tr][br][td]Roots[/td][br][td]根[/td][br][td]근[/td][br][td]根[/td][br][/tr][br][tr][br][td]X-axis Intersection[/td][br][td]X軸との交点[/td][br][td]x축과의 교점[/td][br][td]与x轴的交点[/td][br][/tr][br][tr][br][td]Graph Sketching[/td][br][td]グラフのスケッチ[/td][br][td]그래프 스케치[/td][br][td]绘图[/td][br][/tr][br][tr][br][td]Discriminant[/td][br][td]判別式[/td][br][td]판별식[/td][br][td]判别式[/td][br][/tr][br][tr][br][td]Leading Coefficient[/td][br][td]最高次係数[/td][br][td]선도계수[/td][br][td]首项系数[/td][br][/tr][br][tr][br][td]Orientation of Quadratic Graph[/td][br][td]二次関数のグラフの向き[/td][br][td]이차함수 그래프의 방향[/td][br][td]二次图形的方向[/td][br][/tr][br][tr][br][td]Completing the Square[/td][br][td]平方完成[/td][br][td]완전제곱식[/td][br][td]配方法[/td][br][/tr][br][tr][br][td]Quadratic Formula[/td][br][td]二次方程式の解の公式[/td][br][td]이차방정식의 해 공식[/td][br][td]二次方程的求根公式[/td][br][/tr][br][tr][br][td]Solution Regions[/td][br][td]解の領域[/td][br][td]해의 영역[/td][br][td]解的区域[/td][br][/tr][br][tr][br][td]Quadratic Inequalities[/td][br][td]二次不等式[/td][br][td]이차부등식[/td][br][td]二次不等式[/td][br][/tr][br][tr][br][td]Factored Form[/td][br][td]因数分解された形[/td][br][td]인수분해된 형태[/td][br][td]因式分解形式[/td][br][/tr][br][tr][br][td]Perfect Square[/td][br][td]完全平方[/td][br][td]완전제곱[/td][br][td]完全平方[/td][br][/tr][br][tr][br][td]Critical Values[/td][br][td]臨界値[/td][br][td]임계값[/td][br][td]临界值[/td][br][/tr][br][/table][br]

[table][br][tr][br][td][b]Factual Questions[/b][/td][br][td][b]Conceptual Questions[/b][/td][br][td][b]Debatable Questions[/b][/td][br][/tr][br][tr][br][td]1. How do you factorise a quadratic equation in the form [math]ax^2+bx+c[/math]?[/td][br][td]1. Explain the significance of the discriminant in solving quadratic equations.[/td][br][td]1. Is factoring always the most efficient method for solving quadratic equations? Why or why not?[/td][br][/tr][br][tr][br][td]2. What are the roots of the quadratic equation [math]ax^2+bx+c[/math]?[/td][br][td]2. Discuss how the sign of the leading coefficient affects the orientation of a quadratic graph.[/td][br][td]2. Debate the importance of learning to sketch quadratic graphs by hand in the age of graphing calculators.[/td][br][/tr][br][tr][br][td]3. How do you determine if a quadratic graph will touch or intersect the x-axis?[/td][br][td]3. How does factoring a quadratic equation help in sketching its graph?[/td][br][td]3. Can understanding inequalities and their graphical representations enhance problem-solving skills in real-life situations?[/td][br][/tr][br][tr][br][td]4. Sketch the graph of [math]y=ax^2+bx+c.[/math][/td][br][td]4. Compare and contrast the methods of solving quadratic equations: factoring, completing the square, and using the quadratic formula.[/td][br][td]4. Discuss the statement: "The ability to solve quadratic equations is more crucial than understanding their graphical implications."[/td][br][/tr][br][tr][br][td]5. What is the solution to the inequality[math]ax^2+bx+c>0[/math]?[/td][br][td]5. Explain the concept of "solution regions" in quadratic inequalities.[/td][br][td]5. Evaluate the role of quadratic equations and their applications in higher mathematics and other disciplines.[/td][br][/tr][br][/table][br]

Mini-Investigation: The Quest for Quadratic Mastery

Mini-Investigation: The Quest for Quadratic Mastery[br][br]Welcome, math adventurer, to the Quest for Quadratic Mastery! Today's challenge is to unlock the secrets of factoring quadratics. Prepare your algebraic skills, and let's crack these puzzles![br][br]Complete these quests, and you shall be crowned the Quadratic Master! Remember, each equation is a puzzle waiting to be solved. Enjoy the journey![br]

Explain a method for factorising quadratics with a worked example. (The videos explain a process, do you have a different method?)

Factorising quadratics

Factorising harder quadratics

1. Simple Squares: Look at the quadratic equation [math]x^2-18x+81[/math] = 0. It seems to be a perfect square! How can you identify perfect squares? What does the graph of a perfect square parabola look like?

2. Significance of Signs: How does the sign of the solutions relate to the original equation?

3. Reverse Engineering: Imagine you're a mathematician who loves the number 3. Create a quadratic equation that has 3 and -3 as its solutions. What does your equation look like?

4. Creative Coefficients: Design your own factorable quadratic equation. Challenge a friend to factor it. Can they do it without using the quadratic formula?

5. Quadratic Challenge: Find a quadratic equation that is impossible to factor with whole numbers. What does the graph of such an equation look like?[br]

Factorise the quadratic equation [math]x^2-5x+6[/math].[br]

D) [math](x+2)(x-3)[/math] C) [math](x+3)(x+2)[/math][br] B) [math](x-3)(x-2)[/math][br] A) [math](x-1)(x-6)[/math]

Factorise the quadratic equation [math]x^2+4x+4[/math].[br]

[br]C) [math](x+4)(x+1)[/math][br][br][br] B) [math](x-2)^2[/math][br] A) [math](x+2)^2[/math][br][br] D) [math](x-4)(x-1)[/math][br]

Factorise the quadratic equation [math]x^2-4[/math].

D) [math](x-4)(x+4)[/math] C) [math](x+2)^2[/math][br] B) [math](x+2)(x-2)[/math] A) [math](x-2)^2[/math]

Factorise the quadratic equation [math]x^2-9[/math].[br]

C) [math](x+3)^2[/math][br][br] A) [math](x-3)^2[/math][br][br][br] D) [math](x-9)(x+9)[/math] B) [math](x+3)(x-3)[/math][br][br]

Factorise the quadratic equation [math]x^2+x-6[/math].

C) [math](x-3)(x+2)[/math][br][br] D) [math](x+3)(x-2)[/math] B) [math](x+2)(x-3)[/math][br][br] A) [math](x-2)(x+3)[/math][br][br]

Part 2 - Using factored form to sketch quadratics

Part 3 - Quadratic inequalities

We can also factored form to produce a number line for values of x for a which a quadratic is larger or less than 0. Experiment with point A and B to see how critical value (zeroes of the quadratic) can be used to find intervals for quadratic inequalities.

When it is an inequality, factorising can still be useful to identify critical values. For quadratics we can consider the critical valus (the zeroes of the function) and the concavity of the function. For higher degree polynomials this same approach can