2.5 Quadratic equations (Page 5/14)

Page 5 / 14

Solving a quadratic equation with the quadratic formula

Use the quadratic formula to solve $x^{2} + x + 2 = 0.$

First, we identify the coefficients: $a = 1, b = 1,$ and $c = 2.$

Substitute these values into the quadratic formula.

\begin{matrix} x & = & \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ = & \frac{- (1) \pm \sqrt{{(1)}^{2} - (4) \cdot (1) \cdot (2)}}{2 \cdot 1} \\ = & \frac{- 1 \pm \sqrt{1 - 8}}{2} \\ = & \frac{- 1 \pm \sqrt{- 7}}{2} \\ = & \frac{- 1 \pm i \sqrt{7}}{2} \end{matrix}

The solutions to the equation are $\frac{- 1 + i \sqrt{7}}{2}$ and $\frac{- 1 - i \sqrt{7}}{2}$ or $\frac{- 1}{2} + \frac{i \sqrt{7}}{2}$ and $\frac{- 1}{2} - \frac{i \sqrt{7}}{2} .$

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Try It

Solve the quadratic equation using the quadratic formula: $9 x^{2} + 3 x - 2 = 0.$

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

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The discriminant

The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant , or the expression under the radical, $b^{2} - 4 a c .$ The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. [link] relates the value of the discriminant to the solutions of a quadratic equation.

Value of Discriminant	Results
$b^{2} - 4 a c = 0$	One rational solution (double solution)
$b^{2} - 4 a c > 0,$ perfect square	Two rational solutions
$b^{2} - 4 a c > 0,$ not a perfect square	Two irrational solutions
$b^{2} - 4 a c < 0$	Two complex solutions

A General Note

The discriminant

For $a x^{2} + b x + c = 0,$ where $a,$ $b,$ and $c$ are rational and real numbers, the discriminant is the expression under the radical in the quadratic formula: $b^{2} - 4 a c .$ It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.

Using the discriminant to find the nature of the solutions to a quadratic equation

Use the discriminant to find the nature of the solutions to the following quadratic equations:

$x^{2} + 4 x + 4 = 0$
$8 x^{2} + 14 x + 3 = 0$
$3 x^{2} - 5 x - 2 = 0$
$3 x^{2} - 10 x + 15 = 0$

Calculate the discriminant $b^{2} - 4 a c$ for each equation and state the expected type of solutions.

$x^{2} + 4 x + 4 = 0$

$b^{2} - 4 a c = {(4)}^{2} - 4 (1) (4) = 0.$ There will be one rational double solution.
$8 x^{2} + 14 x + 3 = 0$

$b^{2} - 4 a c = {(14)}^{2} - 4 (8) (3) = 100.$ As $100$ is a perfect square, there will be two rational solutions.
$3 x^{2} - 5 x - 2 = 0$

$b^{2} - 4 a c = {(−5)}^{2} - 4 (3) (−2) = 49.$ As $49$ is a perfect square, there will be two rational solutions.
$3 x^{2} −10 x + 15 = 0$

$b^{2} - 4 a c = {(−10)}^{2} - 4 (3) (15) = −80.$ There will be two complex solutions.

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Using the pythagorean theorem

One of the most famous formulas in mathematics is the Pythagorean Theorem . It is based on a right triangle, and states the relationship among the lengths of the sides as $a^{2} + b^{2} = c^{2},$ where $a$ and $b$ refer to the legs of a right triangle adjacent to the $90°$ angle, and $c$ refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.

We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.

The Pythagorean Theorem is given as

a^{2} + b^{2} = c^{2}

where $a$ and $b$ refer to the legs of a right triangle adjacent to the $90^{\circ}$ angle, and $c$ refers to the hypotenuse, as shown in [link] .

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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