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## Solution by the quadratic formula

It is not always possible to solve a quadratic equation by factorising and sometimes it is lengthy and tedious to solve a quadratic equation by completing the square. In these situations, you can use the quadratic formula that gives the solutions to any quadratic equation.

Consider the general form of the quadratic function:

$f\left(x\right)=a{x}^{2}+bx+c.$

Factor out the $a$ to get:

$f\left(x\right)=a\left({x}^{2}+\frac{b}{a}x+\frac{c}{a}\right).$

Now we need to do some detective work to figure out how to turn [link] into a perfect square plus some extra terms. We know that for a perfect square:

${\left(m+n\right)}^{2}={m}^{2}+2mn+{n}^{2}$

and

${\left(m-n\right)}^{2}={m}^{2}-2mn+{n}^{2}$

The key is the middle term on the right hand side, which is $2×$ the first term $×$ the second term of the left hand side. In [link] , we know that the first term is $x$ so 2 $×$ the second term is $\frac{b}{a}$ . This means that the second term is $\frac{b}{2a}$ . So,

${\left(x+\frac{b}{2a}\right)}^{2}={x}^{2}+2\frac{b}{2a}x+{\left(\frac{b}{2a}\right)}^{2}.$

In general if you add a quantity and subtract the same quantity, nothing has changed. This means if we add and subtract ${\left(\frac{b}{2a}\right)}^{2}$ from the right hand side of [link] we will get:

$\begin{array}{ccc}\hfill f\left(x\right)& =& a\left({x}^{2}+\frac{b}{a}x+\frac{c}{a}\right)\hfill \\ & =& a\left({x}^{2},+,\frac{b}{a},x,+,{\left(\frac{b}{2a}\right)}^{2},-,{\left(\frac{b}{2a}\right)}^{2},+,\frac{c}{a}\right)\hfill \\ & =& a\left({\left[x,+,\left(\frac{b}{2a}\right)\right]}^{2},-,{\left(\frac{b}{2a}\right)}^{2},+,\frac{c}{a}\right)\hfill \\ & =& a\left({\left[x,+,\left(\frac{b}{2a}\right)\right]}^{2}\right)+c-\frac{{b}^{2}}{4a}\hfill \end{array}$

We set $f\left(x\right)=0$ to find its roots, which yields:

$a{\left(x+\frac{b}{2a}\right)}^{2}=\frac{{b}^{2}}{4a}-c$

Now dividing by $a$ and taking the square root of both sides gives the expression

$x+\frac{b}{2a}=±\sqrt{\frac{{b}^{2}}{4{a}^{2}}-\frac{c}{a}}$

Finally, solving for $x$ implies that

$\begin{array}{ccc}\hfill x& =& -\frac{b}{2a}±\sqrt{\frac{{b}^{2}}{4{a}^{2}}-\frac{c}{a}}\hfill \\ & =& -\frac{b}{2a}±\sqrt{\frac{{b}^{2}-4ac}{4{a}^{2}}}\hfill \end{array}$

which can be further simplified to:

$x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$

These are the solutions to the quadratic equation. Notice that there are two solutions in general, but these may not always exists (depending on the sign ofthe expression ${b}^{2}-4ac$ under the square root). These solutions are also called the roots of the quadratic equation.

Find the roots of the function $f\left(x\right)=2{x}^{2}+3x-7$ .

1. The expression cannot be factorised. Therefore, the general quadratic formula must be used.

2. From the equation:

$a=2$
$b=3$
$c=-7$
3. Always write down the formula first and then substitute the values of $a,b$ and $c$ .

$\begin{array}{ccc}\hfill x& =& \frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\hfill \\ & =& \frac{-\left(3\right)±\sqrt{{\left(3\right)}^{2}-4\left(2\right)\left(-7\right)}}{2\left(2\right)}\hfill \\ & =& \frac{-3±\sqrt{65}}{4}\hfill \\ & =& \frac{-3±\sqrt{65}}{4}\hfill \end{array}$
4. The two roots of $f\left(x\right)=2{x}^{2}+3x-7$ are $x=\frac{-3+\sqrt{65}}{4}$ and $\frac{-3-\sqrt{65}}{4}$ .

Find the solutions to the quadratic equation ${x}^{2}-5x+8=0$ .

1. The expression cannot be factorised. Therefore, the general quadratic formula must be used.

2. From the equation:

$a=1$
$b=-5$
$c=8$
3. $\begin{array}{ccc}\hfill x& =& \frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\hfill \\ & =& \frac{-\left(-5\right)±\sqrt{{\left(-5\right)}^{2}-4\left(1\right)\left(8\right)}}{2\left(1\right)}\hfill \\ & =& \frac{5±\sqrt{-7}}{2}\hfill \end{array}$
4. Since the expression under the square root is negative these are not real solutions ( $\sqrt{-7}$ is not a real number). Therefore there are no real solutions to the quadratic equation ${x}^{2}-5x+8=0$ . This means that the graph of the quadratic function $f\left(x\right)={x}^{2}-5x+8$ has no $x$ -intercepts, but that the entire graph lies above the $x$ -axis.

## Solution by the quadratic formula

Solve for $t$ using the quadratic formula.

1. $3{t}^{2}+t-4=0$
2. ${t}^{2}-5t+9=0$
3. $2{t}^{2}+6t+5=0$
4. $4{t}^{2}+2t+2=0$
5. $-3{t}^{2}+5t-8=0$
6. $-5{t}^{2}+3t-3=0$
7. ${t}^{2}-4t+2=0$
8. $9{t}^{2}-7t-9=0$
9. $2{t}^{2}+3t+2=0$
10. ${t}^{2}+t+1=0$
• In all the examples done so far, the solutions were left in surd form. Answers can also be given in decimal form, using the calculator. Read the instructions when answering questions in a test or exam whether to leave answers in surd form, or in decimal form to an appropriate number of decimal places.
• Completing the square as a method to solve a quadratic equation is only done when specifically asked.

## Mixed exercises

Solve the quadratic equations by either factorisation, completing the square or by using the quadratic formula:

• Always try to factorise first, then use the formula if the trinomial cannot be factorised.
• Do some of them by completing the square and then compare answers to those done using the other methods.
 1. $24{y}^{2}+61y-8=0$ 2. $-8{y}^{2}-16y+42=0$ 3. $-9{y}^{2}+24y-12=0$ 4. $-5{y}^{2}+0y+5=0$ 5. $-3{y}^{2}+15y-12=0$ 6. $49{y}^{2}+0y-25=0$ 7. $-12{y}^{2}+66y-72=0$ 8. $-40{y}^{2}+58y-12=0$ 9. $-24{y}^{2}+37y+72=0$ 10. $6{y}^{2}+7y-24=0$ 11. $2{y}^{2}-5y-3=0$ 12. $-18{y}^{2}-55y-25=0$ 13. $-25{y}^{2}+25y-4=0$ 14. $-32{y}^{2}+24y+8=0$ 15. $9{y}^{2}-13y-10=0$ 16. $35{y}^{2}-8y-3=0$ 17. $-81{y}^{2}-99y-18=0$ 18. $14{y}^{2}-81y+81=0$ 19. $-4{y}^{2}-41y-45=0$ 20. $16{y}^{2}+20y-36=0$ 21. $42{y}^{2}+104y+64=0$ 22. $9{y}^{2}-76y+32=0$ 23. $-54{y}^{2}+21y+3=0$ 24. $36{y}^{2}+44y+8=0$ 25. $64{y}^{2}+96y+36=0$ 26. $12{y}^{2}-22y-14=0$ 27. $16{y}^{2}+0y-81=0$ 28. $3{y}^{2}+10y-48=0$ 29. $-4{y}^{2}+8y-3=0$ 30. $-5{y}^{2}-26y+63=0$ 31. ${x}^{2}-70=11$ 32. $2{x}^{2}-30=2$ 33. ${x}^{2}-16=2-{x}^{2}$ 34. $2{y}^{2}-98=0$ 35. $5{y}^{2}-10=115$ 36. $5{y}^{2}-5=19-{y}^{2}$

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Other chapter Q/A we can ask