# 2.5 Quadratic equations  (Page 5/14)

 Page 5 / 14

Use the quadratic formula to solve $\text{\hspace{0.17em}}{x}^{2}+x+2=0.$

First, we identify the coefficients: $\text{\hspace{0.17em}}a=1,b=1,$ and $\text{\hspace{0.17em}}c=2.$

Substitute these values into the quadratic formula.

$\begin{array}{ccc}\hfill x& =& \frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\hfill \\ & =& \frac{-\left(1\right)±\sqrt{{\left(1\right)}^{2}-\left(4\right)\cdot \left(1\right)\cdot \left(2\right)}}{2\cdot 1}\hfill \\ & =& \frac{-1±\sqrt{1-8}}{2}\hfill \\ & =& \frac{-1±\sqrt{-7}}{2}\hfill \\ & =& \frac{-1±i\sqrt{7}}{2}\hfill \end{array}$

The solutions to the equation are $\text{\hspace{0.17em}}\frac{-1+i\sqrt{7}}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{-1-i\sqrt{7}}{2}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\frac{-1}{2}+\frac{i\sqrt{7}}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{-1}{2}-\frac{i\sqrt{7}}{2}.$

Solve the quadratic equation using the quadratic formula: $\text{\hspace{0.17em}}9{x}^{2}+3x-2=0.$

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

## 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, $\text{\hspace{0.17em}}{b}^{2}-4ac.\text{\hspace{0.17em}}$ 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
${b}^{2}-4ac=0$ One rational solution (double solution)
${b}^{2}-4ac>0,$ perfect square Two rational solutions
${b}^{2}-4ac>0,$ not a perfect square Two irrational solutions
${b}^{2}-4ac<0$ Two complex solutions

## The discriminant

For $\text{\hspace{0.17em}}a{x}^{2}+bx+c=0,$ where $\text{\hspace{0.17em}}a,$ $b,$ and $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ are rational and real numbers, the discriminant    is the expression under the radical in the quadratic formula: $\text{\hspace{0.17em}}{b}^{2}-4ac.\text{\hspace{0.17em}}$ 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:

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

Calculate the discriminant $\text{\hspace{0.17em}}{b}^{2}-4ac\text{\hspace{0.17em}}$ for each equation and state the expected type of solutions.

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

${b}^{2}-4ac={\left(4\right)}^{2}-4\left(1\right)\left(4\right)=0.\text{\hspace{0.17em}}$ There will be one rational double solution.

2. $8{x}^{2}+14x+3=0$

${b}^{2}-4ac={\left(14\right)}^{2}-4\left(8\right)\left(3\right)=100.\text{\hspace{0.17em}}$ As $\text{\hspace{0.17em}}100\text{\hspace{0.17em}}$ is a perfect square, there will be two rational solutions.

3. $3{x}^{2}-5x-2=0$

${b}^{2}-4ac={\left(-5\right)}^{2}-4\left(3\right)\left(-2\right)=49.\text{\hspace{0.17em}}$ As $\text{\hspace{0.17em}}49\text{\hspace{0.17em}}$ is a perfect square, there will be two rational solutions.

4. $3{x}^{2}-10x+15=0$

${b}^{2}-4ac={\left(-10\right)}^{2}-4\left(3\right)\left(15\right)=-80.\text{\hspace{0.17em}}$ There will be two complex solutions.

## 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 $\text{\hspace{0.17em}}{a}^{2}+{b}^{2}={c}^{2},$ where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ refer to the legs of a right triangle adjacent to the $\text{\hspace{0.17em}}90°\text{\hspace{0.17em}}$ angle, and $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ refer to the legs of a right triangle adjacent to the $\text{\hspace{0.17em}}{90}^{\circ }\text{\hspace{0.17em}}$ angle, and $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ refers to the hypotenuse, as shown in [link] .

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin