Topics
8. Solving Quadratic Equations
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United States of AmericaFL
Grade 912

8.01 Solving quadratic equations using graphs and tables

Lesson
Practice
Lesson

Concept summary

A quadratic equation is a polynomial equation of degree 2. The standard form of a quadratic equation is written in the in the form ax^2+bx+c=0 where a, b, and c are real numbers.

We can solve some quadratic equations by drawing the graph of the corresponding function. This also allows us to determine the number of real solutions it has.

x
y
One real solution
x
y
Two real solutions
x
y
No real solutions

The solutions to a quadaratic equation are the x-intercepts of the corresponding function. They also known as the roots of the equation or the zeros of the function. A quadratic equation with no real solutions is said to have non-real solutions.

The zeros of an equation can be also be seen in a table of values, provided the right values of x are chosen, and the equation has at least one real solution.

Worked examples

Example 1

Complete a table of values for y=2x^2-18 and then determine the solutions to the corresponding equation 2x^2-18=0.

Approach

When building a table we want to choose values with suitable range so we don't have to do too many calculations. Start by finding the values in the domain -4\leq x\leq4. If the function value is zero for any of these x-values, then we have found a solution to the corresponding equation.

Solution

x-4-3-2-101234
y140-8-16-18-16-8014

We can see the equation has solutions of x=-3,x=3, which we can also write as x= \pm 3.

Reflection

We can see that the table of values has both positive and negative values. Whenever this is the case for a function of the form f\left(x\right)=ax^2+bx+c we know that the equation 0=ax^2+bx+c must have two real solutions, and the corresponding parabola will have two x-intercepts.

Example 2

Consider the function y=\left(x-2\right)^2-9.

a

Draw a graph of the function.

Approach

The function is given in vertex form so we know the vertex is at \left(2, -9\right). We can substitute x=0 to find the y-intercept at \left(0, -5\right). We can find other points on the curve by substituting in other values, and by filling a table of values.

Solution

-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-10
-8
-6
-4
-2
2
4
6
y

Reflection

It is important when drawing graphs to clearly show the key features such as the vertex and the intercepts.

b

Determine the solution to the equation \left(x-2\right)^2=9.

Approach

The solutions to the equation can be found at the x-intercepts of the graph we just drew, as \left(x-2\right)^2-9=0 is an equivalent equation.

Solution

The solutions are x=-1 and x=5.

Reflection

In this case, the equation has integer solutions, which makes solving graphically an effective method. In other cases the answer can be irrational and so solving graphically may only help determine an approximate solution.

Outcomes

MA.912.AR.3.1

Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real number system.

MA.912.AR.3.7

Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.

MA.912.AR.3.8

Solve and graph mathematical and real-world problems that are modeled with quadratic functions. Interpret key features and determine constraints in terms of the context.

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