Grade 9

Intermediate

9 min

Lesson 6 of 12

The Universal Quadratic Formula

Not Started

Learn the ultimate tool for solving any quadratic equation, even those that cannot be easily factored.

How do NASA engineers calculate the exact moment a rocket must fire its engines to land on a moving planet? They use a formula that works every single time, even when the numbers are too 'messy' for simple mental math.

Learning Objectives

1.By the end of this lesson, you will be able to memorize and correctly apply the quadratic formula to find solutions for any quadratic equation.
2.You'll understand how to use the discriminant to predict the number and type of solutions before solving.
3.You will be able to simplify radical expressions found within your final results.

The Master Key to Quadratics

While factoring is a great shortcut, it doesn't work for every equation. The Quadratic Formula is the 'universal tool' that solves any quadratic equation, provided it is written in Standard Form:

ax^2 + bx + c = 0

. In this form,

a

b

, and

c

are just numbers (coefficients). The formula is:

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

The symbol

\pm

(plus-minus) is the secret to finding the two possible points where a parabola might cross the x-axis. Think of the formula as a recipe: as long as you identify your ingredients (

a

b

, and

c

) correctly, the result will always be correct.

Solve for $x$ : $x^2 - 5x + 6 = 0$

1. Identify the coefficients:

a = 1

b = -5

c = 6

. 2. Plug into the formula:

x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)}

3. Simplify the inside:

x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2}

4. Split into two paths:

x = \frac{5 + 1}{2} = 3

and

x = \frac{5 - 1}{2} = 2

. 5. The solutions are

x = 3

and

x = 2

Quick Check

What must you do to the equation $x^2 + 4x = -3$ before you can use the quadratic formula?

Remember the equation must be in Standard Form:

ax^2 + bx + c = 0

Answer

You must move the -3 to the left side by adding 3 to both sides to set the equation to zero ( $x^2 + 4x + 3 = 0$ ).

The Discriminant: Predicting the Future

Before doing the full calculation, you can look into a 'mathematical crystal ball' called the discriminant. This is the part of the formula located under the square root:

D = b^2 - 4ac

. The value of

D

tells you how many solutions to expect: - If

D > 0

: You get two real solutions (the graph crosses the x-axis twice). - If

D = 0

: You get one real solution (the graph's vertex touches the x-axis). - If

D < 0

: You get no real solutions (the graph never touches the x-axis).

Predict the number of solutions for $2x^2 + 4x + 5 = 0$ .

1. Identify

a=2, b=4, c=5

. 2. Calculate

D = b^2 - 4ac

D = (4)^2 - 4(2)(5)

3. Simplify:

D = 16 - 40 = -24

. 4. Since

-24

is less than zero, there are no real solutions.

Quick Check

If your discriminant is exactly 0, how many times does the parabola touch the x-axis?

Answer

Exactly once.

Simplifying Messy Radicals

In the real world, solutions are rarely clean whole numbers. Often, you will end up with a radical (square root) that cannot be perfectly solved. To keep your answer exact, you must simplify the radical rather than using a decimal. For example, $\sqrt{20}$ should be simplified to $2\sqrt{5}$ . This ensures your mathematical 'blueprint' remains perfectly accurate without rounding errors.

Solve $x^2 - 4x + 1 = 0$ :

1. $a=1, b=-4, c=1$ . 2. $x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)}$ 3. $x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2}$ 4. Simplify $\sqrt{12}$ to $2\sqrt{3}$ : $x = \frac{4 \pm 2\sqrt{3}}{2}$ 5. Divide both terms in the numerator by 2: $x = 2 \pm \sqrt{3}$ .

Key Takeaways

1The Quadratic Formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves any quadratic equation in standard form.
2The discriminant ( $b^2 - 4ac$ ) determines if there are 2, 1, or 0 real solutions.
3Always simplify radicals and fractions to provide the most accurate 'exact' answer.
4Standard form ( $ax^2 + bx + c = 0$ ) is required before you can identify $a$ , $b$ , and $c$ .

Quick Check

Multiple Choice

In the equation $5 - 2x + x^2 = 0$ , what is the value of $b$ ?

Multiple Choice

What does a discriminant of $16$ indicate?

True/False

The quadratic formula can be used to solve an equation even if it cannot be factored into whole numbers.

What's Next?

Review Tomorrow

In 24 hours, try to write the Quadratic Formula from memory on a blank piece of paper and check it against your notes.

Practice Activity

Find a quadratic equation from your textbook and calculate only the discriminant. Predict the number of solutions, then solve it fully to see if you were right.

Browse

Browse

The Universal Quadratic Formula

Learning Objectives

The Master Key to Quadratics

The Discriminant: Predicting the Future

Simplifying Messy Radicals

Key Takeaways

Quick Check

What's Next?

The Universal Quadratic Formula

Learning Objectives

The Master Key to Quadratics

The Discriminant: Predicting the Future

Simplifying Messy Radicals

Key Takeaways

Quick Check

What's Next?