10.3 Solve quadratic equations using the quadratic formula (Page 2/5)

Elementary algebra Page 2 / 5

Solve $x^{2} - 6 x + 5 = 0$ by using the Quadratic Formula.

Solution


This equation is in standard form.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c.
Simplify.
Rewrite to show two solutions.
Simplify.
Check.

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Solve $a^{2} - 2 a - 15 = 0$ by using the Quadratic Formula.

$a = −3, a = 5$

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Solve $b^{2} + 10 b + 24 = 0$ by using the Quadratic Formula.

$b = −6, b = −4$

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When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the Quadratic Formula. If we get a radical as a solution, the final answer must have the radical in its simplified form.

Solve $4 y^{2} - 5 y - 3 = 0$ by using the Quadratic Formula.

Solution

We can use the Quadratic Formula to solve for the variable in a quadratic equation, whether or not it is named ‘ x ’.


This equation is in standard form.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c.
Simplify.
Rewrite to show two solutions.
Check. We leave the check to you.

Got questions? Get instant answers now!

Solve $2 p^{2} + 8 p + 5 = 0$ by using the Quadratic Formula.

$p = \frac{−4 \pm \sqrt{6}}{2}$

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Solve $5 q^{2} - 11 q + 3 = 0$ by using the Quadratic Formula.

$q = \frac{11 \pm \sqrt{61}}{10}$

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Solve $2 x^{2} + 10 x + 11 = 0$ by using the Quadratic Formula.

Solution


This equation is in standard form.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c.
Simplify.
Simplify the radical.
Factor out the common factor in the numerator.
Remove the common factors.
Rewrite to show two solutions.
Check. We leave the check to you.

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Solve $3 m^{2} + 12 m + 7 = 0$ by using the Quadratic Formula.

$m = \frac{−6 \pm \sqrt{15}}{3}$

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Solve $5 n^{2} + 4 n - 4 = 0$ by using the Quadratic Formula.

$n = \frac{−2 \pm 2 \sqrt{6}}{5}$

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We cannot take the square root of a negative number. So, when we substitute $a$ , $b$ , and $c$ into the Quadratic Formula, if the quantity inside the radical is negative, the quadratic equation has no real solution. We will see this in the next example.

Solve $3 p^{2} + 2 p + 9 = 0$ by using the Quadratic Formula.

Solution

This equation is in standard form.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c.
Simplify.
Simplify the radical.
We cannot take the square root of a negative number.	There is no real solution.

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Solve $4 a^{2} - 3 a + 8 = 0$ by using the Quadratic Formula.

no real solution

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Solve $5 b^{2} + 2 b + 4 = 0$ by using the Quadratic Formula.

no real solution

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The quadratic equations we have solved so far in this section were all written in standard form, $a x^{2} + b x + c = 0$ . Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.

Solve $x (x + 6) + 4 = 0$ by using the Quadratic Formula.

Solution


Distribute to get the equation in standard form.
This equation is now in standard form.
Identify the a, b, c values.
Write the Quadratic Formula.
Then substitute in the values of a, b, c.
Simplify.
Simplify inside the radical.
Simplify the radical.
Factor out the common factor in the numerator.
Remove the common factors.
Rewrite to show two solutions.
Check. We leave the check to you.

Got questions? Get instant answers now!

Solve $x (x + 2) - 5 = 0$ by using the Quadratic Formula.

$x = −1 \pm \sqrt{6}$

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Solve $y (3 y - 1) - 2 = 0$ by using the Quadratic Formula.

$y = - \frac{2}{3}, y = 1$

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Questions & Answers

what is phylogeny

Odigie Reply

evolutionary history and relationship of an organism or group of organisms

AI-Robot

Deng

what is biology

Hajah Reply

the study of living organisms and their interactions with one another and their environments

AI-Robot

what is biology

Victoria Reply

HOW CAN MAN ORGAN FUNCTION

Alfred Reply

the diagram of the digestive system

Assiatu Reply

allimentary cannel

Ogenrwot

How does twins formed

William Reply

They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together

Oluwatobi

what is genetics

Josephine Reply

Genetics is the study of heredity

Misack

how does twins formed?

Misack

What is manual

Hassan Reply

discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles

Joseph Reply

what is biology

Yousuf Reply

the study of living organisms and their interactions with one another and their environment.

Wine

discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form

Joseph Reply

what is the blood cells

Shaker Reply

list any five characteristics of the blood cells

Shaker

lack electricity and its more savely than electronic microscope because its naturally by using of light

Abdullahi Reply

advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear

Abdullahi

cell theory state that every organisms composed of one or more cell,cell is the basic unit of life

Abdullahi

is like gone fail us

DENG

cells is the basic structure and functions of all living things

Ramadan

What is classification

ISCONT Reply

is organisms that are similar into groups called tara

Yamosa

in what situation (s) would be the use of a scanning electron microscope be ideal and why?

Kenna Reply

A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,

Hilary

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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