155e. Solving Simple Quadratic Equations

Learning Intentions

• To know the form of a simple quadratic equation
• Determine the number of solutions to a simple quadratic equation
• Solve a simple quadratic equation

Pre-requisite Summary

• An equation is a statement that two expressions are equal
• A solution to an equation is a value that makes the equation true
• A quadratic expression contains a term with a variable squared, such as
• A simple quadratic equation can often be written in the form
• If , then solving the equation means finding numbers whose square is
• A positive number has two square roots, has one square root, and a negative number has no real square roots
• Solutions should be checked by substitution into the original equation

Worked Examples

Worked Example 1

State whether each equation is quadratic.

a)

b)

c)

Worked Example 2

For each equation, determine the number of real solutions.

a)

b)

c)

Worked Example 3

Solve .

Worked Example 4

Solve .

Worked Example 5

Solve .

Worked Example 6

Solve .

Problems

Problem 1

State whether each equation is quadratic.

a)

b)

c)

Problem 2

For each equation, determine the number of real solutions.

a)

b)

c)

Problem 3

Solve .

Problem 4

Solve .

Problem 5

Solve .

Problem 6

Solve .

Exercises

Understanding and Fluency

Exercise 1.

Identify whether each equation is quadratic.

a)

b)

c)

Exercise 2.

Identify whether each equation is quadratic.

a)

b)

c)

Exercise 3.

Determine the number of real solutions to each equation.

a)

b)

c)

Exercise 4.

Determine the number of real solutions to each equation.

a)

b)

c)

Exercise 5.

Solve each equation.

a)

b)

c)

Exercise 6.

Solve each equation.

a)

b)

c)

Exercise 7.

Solve each equation.

a)

b)

c)

Exercise 8.

Solve each equation.

a)

b)

c)

Exercise 9.

Solve and Check each equation.

a)

b)

Exercise 10.

Solve and check each equation.

a)

b)

Reasoning

Exercise 11.

Explain why has two real solutions but has only one real solution.

Exercise 12.

A student says that the solutions to are only . Explain the error.

Exercise 13.

Noah says that has two solutions, and . Explain why Noah is incorrect in the real numbers.

Exercise 14.

Explain why solving begins by rewriting the equation as .

Exercise 15.

A student says that every quadratic equation has two real solutions. Give an example to show why this is false.

Problem-solving

Exercise 16.

The area of a square is . Let the side length be cm. Write and solve a quadratic equation for the side length.

Exercise 17.

A square garden has area . Let the side length be m. Write and solve a quadratic equation for .

Exercise 18.

The side length of a square is represented by . If the area is , write and solve a quadratic equation to Solve the possible values of , then state which value makes sense for the context.

Exercise 19.

A physics model gives the equation for a time value . Solve the equation and state how many real solutions it has.

Exercise 20.

A student solves and writes . Complete the solution and explain why there is another real solution.

Potential Misunderstandings

• Thinking any equation with a pronumeral is quadratic
• Thinking a quadratic equation must always be written exactly as before it can be recognised
• Confusing a squared variable, such as , with a variable multiplied by
• Believing that every quadratic equation has exactly two real solutions
• Forgetting that has only one real solution
• Thinking a negative number can be the square of a real number
• Giving only the positive square root when solving equations such as
• Forgetting to first isolate the squared term before taking square roots
• Treating as if the solution is only
• Failing to check solutions by substitution into the original equation

Next: 156. Formulas