Lesson Note
INTRODUCTION
Quadratic equations are the backbone of many areas in mathematics and everyday problem solving. They describe relationships where the unknown variable appears raised to the second power and they model a wide range of real‑world situations such as projectile motion, economics, and engineering design. In this lesson we will move beyond the simple techniques of factoring and the quadratic formula to explore deeper properties of quadratic equations, how to transform them into useful forms, and how to read and draw their graphs. By mastering these concepts, students will be able to tackle more complex problems and see the relevance of algebra in the world around them.
MAIN CONTENT
Standard Form and Coefficients
In the standard form a quadratic equation is written as
where $a$, $b$, and $c$ are constants and $a\neq0$. The coefficient $a$ is called the leading coefficient, $b$ is the linear coefficient, and $c$ is the constant term. The values of these coefficients determine the shape and position of the parabola represented by the equation.
Example – Solving by Factoring
Step 1: Write the equation in standard form: $x^{2}-5x+6=0$.
Step 2: Find two numbers that multiply to $c=6$ and add to $b=-5$. These numbers are $-2$ and $-3$.
Step 3: Factor the quadratic: $(x-2)(x-3)=0$.
Step 4: Set each factor equal to zero: $x-2=0$ or $x-3=0$.
Step 5: Solve for $x$: $x=2$ or $x=3$.
Practice Question 1: Solve $2x^{2}+3x-5=0$ by factoring.
Answer 1: $x=1$ or $x=-\dfrac{5}{2}$.
Practice Question 2: Identify the coefficients $a$, $b$, and $c$ in $-4x^{2}+7x-2=0$.
Answer 2: $a=-4$, $b=7$, $c=-2$.
Practice Question 3: State the condition that must be satisfied by $a$ in a quadratic equation.
Answer 3: $a$ must not be zero.
Discriminant and Nature of Roots
The discriminant of a quadratic equation $ax^{2}+bx+c=0$ is given by
It tells us the number and type of roots without solving the equation.
If $\Delta>0$, there are two distinct real roots.
If $\Delta=0$, there is one repeated real root.
If $\Delta<0$, the roots are complex conjugates.
Example – Determining Roots from the Discriminant
Step 1: Consider $3x^{2}-12x+9=0$.
Step 2: Compute $\Delta = (-12)^{2}-4(3)(9)=144-108=36$.
Step 3: Since $\Delta>0$, the equation has two distinct real roots.
Step 4: Use the quadratic formula: $$x=\dfrac{-b\pm\sqrt{\Delta}}{2a}=\dfrac{12\pm6}{6}$$.
Step 5: Simplify: $x=3$ or $x=1$.
Practice Question 1: Find $\Delta$ for $x^{2}+4x+3=0$.
Answer 1: $\Delta=4$.
Practice Question 2: How many real roots does $x^{2}+2x+5=0$ have?
Answer 2: None (complex roots).
Practice Question 3: If $\Delta=0$ for $ax^{2}+bx+c=0$, what is the root?
Answer 3: $x=-\dfrac{b}{2a}$.
Vertex Form and Completing the Square
The vertex form of a quadratic function is
where $(h,k)$ is the vertex of the parabola. To convert $ax^{2}+bx+c$ to this form we complete the square.
Example – Converting to Vertex Form
Step 1: Start with $y=2x^{2}+8x+5$.
Step 2: Factor out the leading coefficient from the $x$ terms: $y=2(x^{2}+4x)+5$.
Step 3: Complete the square inside: $x^{2}+4x+4-4 = (x+2)^{2}-4$.
Step 4: Substitute back: $y=2[(x+2)^{2}-4]+5 = 2(x+2)^{2}-8+5$.
Step 5: Simplify: $y=2(x+2)^{2}-3$.
Thus the vertex is $(-2,-3)$.
Practice Question 1: Convert $y=x^{2}-6x+8$ to vertex form.
Answer 1: $y=(x-3)^{2}-1$.
Practice Question 2: Identify the vertex of $y=3(x+1)^{2}+4$.
Answer 2: $(-1,4)$.
Practice Question 3: What is the axis of symmetry for $y=5x^{2}+10x+1$?
Answer 3: $x=-1$.
Graphing Quadratic Functions
A quadratic function $y=ax^{2}+bx+c$ traces a parabola. The leading coefficient $a$ determines whether it opens upward ($a>0$) or downward ($a<0$). The vertex $(h,k)$ is the turning point. The axis of symmetry is the vertical line $x=h$. The $x$‑intercepts are the real roots (if any) found by solving $ax^{2}+bx+c=0$. The $y$‑intercept is the value of $c$.
Example – Sketching $y=-x^{2}+4x-3$
Step 1: Identify $a=-1$, $b=4$, $c=-3$.
Step 2: Find the vertex using $h=-\dfrac{b}{2a}=\dfrac{4}{2}=2$, $$k=c-\dfrac{b^{2}}{4a}=-3-\dfrac{16}{-4}=1$$.
Step 3: Since $a<0$, the parabola opens downward.
Step 4: Compute $x$‑intercepts by solving $-x^{2}+4x-3=0$: $(x-1)(x-3)=0$.
Step 5: $x$‑intercepts are $x=1$ and $x=3$.
Step 6: $y$‑intercept is $-3$.
Step 7: Plot the points $(2,1)$, $(1,0)$, $(3,0)$, $(0,-3)$ and sketch the parabola symmetric about $x=2$.
Practice Question 1: Determine the direction of opening for $y=7x^{2}-5$.
Answer 1: Upward.
Practice Question 2: Find the $y$‑intercept of $y=-2x^{2}+3x+5$.
Answer 2: $5$.
Practice Question 3: Identify the axis of symmetry for $y=4(x-2)^{2}+1$.
Answer 3: $x=2$.
Applications of Quadratic Equations
Quadratic equations model many real‑world scenarios. For example, the height $h$ of a projectile launched with initial velocity $v_0$ at an angle $\theta$ after time $t$ is given by
where $g$ is the acceleration due to gravity. Setting $h=0$ yields a quadratic equation in $t$ that gives the time of flight.
Example – Time of Flight of a Ball
Step 1: Suppose a ball is thrown upward with $v_{0}=20$ m/s and $g=10$ m/s$^{2}$.
Step 2: The height equation is $h = -5t^{2}+20t$.
Step 3: Set $h=0$: $-5t^{2}+20t=0$.
Step 4: Factor: $-5t(t-4)=0$.
Step 5: Solve: $t=0$ (launch) or $t=4$ seconds (ball hits ground).
Practice Question 1: A car accelerates from rest with $a=3$ m/s$^{2}$ for $t$ seconds. Its distance $s$ satisfies $s= \dfrac{1}{2}at^{2}$. Find $t$ when $s=45$ m.
Answer 1: $t=5$ s.
Practice Question 2: A bridge has a parabolic arch described by $y= -0.5x^{2}+4x$. What is the maximum height?
Answer 2: $8$ units.
Practice Question 3: Solve $x^{2}-10x+25=0$ and interpret the result in terms of a real‑world scenario.
Answer 3: $x=5$ (double root, e.g., a projectile that lands immediately).
SUMMARY
Quadratic equations, though seemingly simple, possess rich structure that can be explored through discriminants, vertex forms, and graphing techniques. By mastering the standard form, completing the square, and understanding the geometric significance of the vertex and axis of symmetry, students can visualize solutions and apply these concepts to real‑world problems such as projectile motion and optimization. The ability to switch between algebraic and graphical interpretations empowers students to solve problems efficiently and to appreciate the interconnectedness of mathematical ideas.
EVALUATION
1. Find the discriminant of $5x^{2}-20x+15=0$.
2. Convert $y=4x^{2}-8x+3$ to vertex form.
3. Determine the axis of symmetry for $y=-2x^{2}+12x-10$.
4. Solve $x^{2}+6x+9=0$ using the quadratic formula.
5. If $y=3(x-1)^{2}+2$, what is the $y$‑intercept?