155e. Solving Simple Quadratic Equations

Learning Intentions

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 $x^{2}$
A simple quadratic equation can often be written in the form $x^{2} = k$
If $x^{2} = k$ , then solving the equation means finding numbers whose square is $k$
A positive number has two square roots, $0$ has one square root, and a negative number has no real square roots
Solutions should be checked by substitution into the original equation

State whether each equation is quadratic.
a) $x^{2} = 16$
b) $3 x = 12$
c) $y^{2} + 5 = 9$

For each equation, determine the number of real solutions.
a) $x^{2} = 25$
b) $x^{2} = 0$
c) $x^{2} = - 4$

Solve $x^{2} = 36$ .

Solve $m^{2} = 49$ .

Solve $p^{2} - 9 = 0$ .

Solve $a^{2} + 7 = 32$ .

State whether each equation is quadratic.
a) $x^{2} = 9$
b) $2 x + 1 = 7$
c) $t^{2} - 4 = 0$

For each equation, determine the number of real solutions.
a) $x^{2} = 64$
b) $x^{2} = 0$
c) $x^{2} = - 9$

Solve $x^{2} = 81$ .

Solve $m^{2} = 64$ .

Solve $p^{2} - 16 = 0$ .

Solve $a^{2} + 5 = 41$ .

Identify whether each equation is quadratic.
a) $x^{2} = 4$
b) $x + 2 = 9$
c) $3 y^{2} = 27$
Identify whether each equation is quadratic.
a) $m^{2} - 1 = 0$
b) $2 p = 10$
c) $a^{2} + 6 = 15$
Determine the number of real solutions to each equation.
a) $x^{2} = 16$
b) $x^{2} = 0$
c) $x^{2} = - 1$
Determine the number of real solutions to each equation.
a) $m^{2} = 100$
b) $p^{2} = - 25$
c) $a^{2} = 0$
Solve each equation.
a) $x^{2} = 1$
b) $x^{2} = 49$
c) $x^{2} = 121$
Solve each equation.
a) $m^{2} = 9$
b) $p^{2} = 144$
c) $t^{2} = 169$
Solve each equation.
a) $x^{2} - 4 = 0$
b) $y^{2} - 25 = 0$
c) $a^{2} - 64 = 0$
Solve each equation.
a) $m^{2} + 3 = 28$
b) $p^{2} + 8 = 57$
c) $k^{2} + 1 = 82$
Solve and check each equation.
a) $x^{2} = 36$
b) $y^{2} - 49 = 0$
Solve and check each equation.
a) $a^{2} + 4 = 53$
b) $m^{2} = 0$

Explain why $x^{2} = 25$ has two real solutions but $x^{2} = 0$ has only one real solution.
A student says that the solutions to $x^{2} = 16$ are only $x = 4$ . Explain the error.
Noah says that $x^{2} = - 9$ has two solutions, $x = 3$ and $x = - 3$ . Explain why Noah is incorrect in the real numbers.
Explain why solving $p^{2} - 9 = 0$ begins by rewriting the equation as $p^{2} = 9$ .
A student says that every quadratic equation has two real solutions. Give an example to show why this is false.

The area of a square is $64 {cm}^{2}$ . Let the side length be $s$ cm. Write and solve a quadratic equation for the side length.
A square garden has area $121 m^{2}$ . Let the side length be $x$ m. Write and solve a quadratic equation for $x$ .
The side length of a square is represented by $p$ . If the area is $36 {cm}^{2}$ , write and solve a quadratic equation to find the possible values of $p$ , then state which value makes sense for the context.
A physics model gives the equation $t^{2} = 0$ for a time value $t$ . Solve the equation and state how many real solutions it has.
A student solves $x^{2} - 100 = 0$ and writes $x = 10$ . Complete the solution and explain why there is another real solution.

Thinking any equation with a pronumeral is quadratic
Thinking a quadratic equation must always be written exactly as $a x^{2} + b x + c = 0$ before it can be recognised
Confusing a squared variable, such as $x^{2}$ , with a variable multiplied by $2$
Believing that every quadratic equation has exactly two real solutions
Forgetting that $x^{2} = 0$ 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 $x^{2} = 49$
Forgetting to first isolate the squared term before taking square roots
Treating $x^{2} = 36$ as if the solution is $x = 6$ only
Failing to check solutions by substitution into the original equation