Explore and apply the quadratic formula using technology

Ontario Grade 11 Mathematics

Quadratic Functions

Find and check the solutions of quadratic equations

A quadratic equation can have two real solutions, one real solution, or no real solutions. Some quadratic equations are difficult to solve by factoring. The quadratic formula gives a method that works for any quadratic equation, and technology can help with the calculations. In this lesson, you will practise entering the values carefully, interpreting the results, and checking them.

What you will learn

Recognize the coefficients in a quadratic equation written in standard form.
Use the quadratic formula with a calculator or graphing tool to find solutions.
Connect equation solutions to the places where a graph crosses the horizontal axis.
Check whether calculator results are reasonable.

1. Prerequisite bridge: identify the parts

A quadratic equation is an equation whose highest variable power is two. For example, an equation such as

2x^2+5x-3=0

is quadratic because its highest power of

x

is

2

.

Before using the formula, write the equation in standard form. Standard form means that the terms are arranged as

ax^2+bx+c=0

. The letters

a

,

b

, and

c

stand for numbers called coefficients. The coefficient of

x^2

is

a

, the coefficient of

x

is

b

, and the constant term is

c

.

Keep the signs with the coefficients. In

2x^2+5x-3=0

, the values are

a=2

,

b=5

, and

c=-3

. A missing coefficient of

x

is zero. For example, in

x^2-9=0

, the coefficient of

x

is

b=0

.

The solutions are the values of

x

that make the equation true. They are also called roots or zeros. A graph can help show them: for the related quadratic graph, real solutions are the

x

-coordinates where the graph meets the horizontal axis.

ax^2+bx+c=0

First rearrange the equation so one side is zero.
Read each coefficient with its sign.
A missing term has coefficient zero.

2. The formula and what technology does

For an equation in standard form, the quadratic formula gives its solution or solutions. Substitute the values of

a

,

b

, and

c

exactly as they appear in the equation. Use brackets around negative values when entering them into a calculator.

The expression under the square root is called the discriminant. It helps describe the number of real solutions. A positive discriminant gives two real solutions, a zero discriminant gives one repeated real solution, and a negative discriminant gives no real solutions. This lesson focuses on using technology to calculate and interpret real solutions.

A calculator can evaluate the formula directly. A graphing tool can also display the quadratic and show where it crosses the horizontal axis. These are two representations of the same solutions: one comes from calculation, and the other comes from the graph.

Use technology to support careful reasoning, not to replace it. Check that the equation is in standard form, enter each coefficient with the correct sign, and compare the results with the graph or a substitution check.

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Substitute the coefficients into the formula without changing their signs.
Use parentheses when entering a negative coefficient.
A graph's horizontal-axis crossings should agree with real equation solutions.

3. Guided example: calculate, view, and check

Consider

x^2-3x-2=0

. It is already in standard form, so

a=1

,

b=-3

, and

c=-2

. We will use a calculator to evaluate both values from the formula.

On a calculator with a general calculation screen, enter the formula twice: once using the plus sign and once using the minus sign. Enter the coefficients as

1

,

-3

, and

-2

. Keep the entire numerator grouped, and divide by

2a

.

A graphing tool offers a visual check. Graph

y=x^2-3x-2

and look for its horizontal-axis crossings. The crossings should be close to the calculator's two answers. The graph is not expected to show many decimal places, so use the calculator values for a more precise result.

To check a decimal answer, substitute it into the original left side. A result close to zero supports the answer. Small nonzero results can occur because displayed decimals are rounded.

The negative sign in $b=-3$ stays attached to $b$ during substitution.
The plus-minus symbol means calculate two values.
Graphing and substitution provide useful checks.

4. Choosing a sensible check

After using technology, ask whether the answers make sense. If the graph appears to cross the horizontal axis twice, two different real answers are reasonable. If your calculation gives only one answer, recheck the formula entry and the plus-minus step.

A common source of error is entering

b

as positive when it is negative. Another is forgetting that the denominator is

2a

, not simply

2

. Brackets make calculator entries clearer, especially when a coefficient is negative.

When checking by substitution, use the original equation. For a proposed solution, replace every

x

with that value and evaluate the left side. It should be zero, or very close to zero if the solution was rounded.

Technology can display a long decimal. Report a suitable rounded value when a decimal is useful, and keep enough digits during checking to avoid creating a large rounding error. If an exact form is displayed or required, retain the square root form as well.

Compare calculator results with the graph's crossings.
Recheck signs, grouping, and the denominator if results disagree.
Substitution into the original equation checks a proposed solution.

From equation to calculator entry

Equation	a	b	c
$x^2-3x-2=0$	$1$	$-3$	$-2$
$2x^2+5x-3=0$	$2$	$5$	$-3$
$x^2-9=0$	$1$	$0$	$-9$

Worked example

Solve with a calculator and verify

Use technology to solve

x^2-3x-2=0

. Give decimal answers to three decimal places, then check the results.

Identify the coefficients
The equation is in standard form. Read each coefficient, including its sign, before entering values into the formula.
$a=1,\quad b=-3,\quad c=-2$
Substitute into the formula
Use the formula with the identified coefficients. The plus-minus symbol calls for two calculations, one with a plus and one with a minus.
$x=\frac{-(-3)\pm\sqrt{(-3)^2-4(1)(-2)}}{2(1)}$
Evaluate both values
Enter each version into a calculator, keeping brackets around negative numbers. The square root is of $17$ , and the denominator is $2$ .
$x=\frac{3\pm\sqrt{17}}{2}\approx 3.562,\ -0.562$
Check the results
Substitute each rounded value into the left side of the original equation. Each result is close to zero; the small difference comes from rounding. A graph of the quadratic should also cross the horizontal axis near both values.
$(3.562)^2-3(3.562)-2\approx 0.001,\quad (-0.562)^2-3(-0.562)-2\approx -0.001$

Answer: The solutions to three decimal places are approximately

x=3.562

and

x=-0.562

.

Check: Both values make the original left side close to zero, and a graph should show horizontal-axis crossings near these values.

Common mistakes and how to avoid them

Using

b=3

for

x^2-3x-2=0

.

Correction: The coefficient includes its sign, so

b=-3

. Use brackets when substituting it.

Calculating only the plus version of the formula.

Correction: The plus-minus symbol represents two calculations. Evaluate once with plus and once with minus.

Dividing by

2

instead of

2a

.

Correction: The full denominator is

2a

. Substitute the value of

a

into that denominator.

Assuming a slightly nonzero substitution result proves a rounded answer is wrong.

Correction: Rounded decimals may not produce exactly zero. Keep more digits for the check and look for a result close to zero.

Lesson summary

Write the quadratic equation in standard form before identifying its coefficients.
Substitute $a$ , $b$ , and $c$ carefully into the quadratic formula.
Use technology to evaluate both possibilities and report sensible decimal values.
Check real solutions with a graph or by substitution into the original equation.

Check your understanding

Question 1

For

2x^2+7x-4=0

, which values should be entered for

a

,

b

, and

c

?

$a=2,\ b=7,\ c=-4$
$a=2,\ b=-7,\ c=4$
$a=7,\ b=2,\ c=-4$
$a=2,\ b=7,\ c=4$

Show answer and explanation

a=2,\ b=7,\ c=-4

The equation is already in standard form. Each coefficient is read with its sign.

Question 2

A calculator gives two values from the plus and minus versions of the formula. What is a useful technology-based check?

Graph the quadratic and see whether its horizontal-axis crossings are near the values.
Change the sign of every coefficient and calculate again.
Use only the larger value because the formula gives two choices.
Check the values in a different quadratic equation.

Show answer and explanation

Graph the quadratic and see whether its horizontal-axis crossings are near the values.

The real solutions are the horizontal-axis crossings of the related graph, so the graph should agree approximately with the calculated values.

Key terms

Quadratic equation: An equation whose highest power of the variable is two.
Standard form: The arrangement $ax^2+bx+c=0$ , where $a$ , $b$ , and $c$ are coefficients.
Coefficient: A number multiplying a term containing a variable.
Solution or root: A value of the variable that makes an equation true.
Discriminant: The expression $b^2-4ac$ in the quadratic formula; its value helps indicate how many real solutions there are.

Continue through MCF3M

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A1.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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