5.2 Solving quadratic equations (Page 2/2)

Siyavula textbooks: grade 10 Page 2 / 2

You can combine these in many ways and so the best way to develop your intuition for the best thing to do is practice problems. A combined set of operations could be, for example,

\begin{matrix} \frac{1}{\sqrt{a x^{2} + b x}} & = & c \\ {(\frac{1}{a x^{2} + b x})}^{- 1} & = & {(c)}^{- 1} & (invert both sides) \\ \frac{\sqrt{a x^{2} + b x}}{1} & = & \frac{1}{c} \\ \sqrt{a x^{2} + b x} & = & \frac{1}{c} \\ {(\sqrt{a x^{2} + b x})}^{2} & = & {(\frac{1}{c})}^{2} & (square both sides) \\ a x^{2} + b x & = & \frac{1}{c^{2}} \end{matrix}

Solve for $x$ : $\sqrt{x + 2} = x$

Square both sides of the equation
Both sides of the equation should be squared to remove the square root sign.

$x + 2 = x^{2}$
Write equation in the form $a x^{2} + b x + c = 0$
$\begin{matrix} x + 2 & = & x^{2} & (subtract x^{2} from both sides) \\ x + 2 - x^{2} & = & 0 & (divide both sides by - 1) \\ - x - 2 + x^{2} & = & 0 \\ x^{2} - x + 2 & = & 0 \end{matrix}$
Factorise the quadratic
$x^{2} - x + 2$

The factors of $x^{2} - x + 2$ are $(x - 2) (x + 1)$ .
Write the equation with the factors
$(x - 2) (x + 1) = 0$
Determine the two solutions
We have

$x + 1 = 0$

or

$x - 2 = 0$

Therefore, $x = - 1$ or $x = 2$ .
Check whether solutions are valid
Substitute $x = - 1$ into the original equation $\sqrt{x + 2} = x$ :

$\begin{matrix} LHS & = & \sqrt{(- 1) + 2} \\ = & \sqrt{1} \\ = & 1 \\ but \\ RHS & = & (- 1) \end{matrix}$

Therefore LHS $â‰$ RHS. The sides of an equation must always balance, a potential solution that does not balance the equation is not valid. In this case the equation does not balance.

Therefore $x â‰ - 1$ .

Now substitute $x = 2$ into original equation $\sqrt{x + 2} = x$ :

$\begin{matrix} LHS & = & \sqrt{2 + 2} \\ = & \sqrt{4} \\ = & 2 \\ and \\ RHS & = & 2 \end{matrix}$

Therefore LHS = RHS

Therefore $x = 2$ is the only valid solution
Write the final answer
$\sqrt{x + 2} = x$ for $x = 2$ only.

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Solve the equation: $x^{2} + 3 x - 4 = 0$ .

Check if the equation is in the form $a x^{2} + b x + c = 0$
The equation is in the required form, with $a = 1$ .
Factorise the quadratic
You need the factors of 1 and 4 so that the middle term is $+ 3$ So the factors are:

$(x - 1) (x + 4)$
Solve the quadratic equation
$x^{2} + 3 x - 4 = (x - 1) (x + 4) = 0$

Therefore $x = 1$ or $x = - 4$ .
Check the solutions
$1^{2} + 3 (1) - 4 = 0$

${(- 4)}^{2} + 3 (- 4) - 4 = 0$

Both solutions are valid.
Write the final solution
Therefore the solutions are $x = 1$ or $x = - 4$ .

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Find the roots of the quadratic equation $0 = - 2 x^{2} + 4 x - 2$ .

Determine whether the equation is in the form $a x^{2} + b x + c = 0$ , with no common factors.
There is a common factor: -2. Therefore, divide both sides of the equation by -2.

$\begin{matrix} - 2 x^{2} + 4 x - 2 & = & 0 \\ x^{2} - 2 x + 1 & = & 0 \end{matrix}$
Factorise $x^{2} - 2 x + 1$
The middle term is negative. Therefore, the factors are $(x - 1) (x - 1)$

If we multiply out $(x - 1) (x - 1)$ , we get $x^{2} - 2 x + 1$ .
Solve the quadratic equation
$x^{2} - 2 x + 1 = (x - 1) (x - 1) = 0$

In this case, the quadratic is a perfect square, so there is only one solution for $x$ : $x = 1$ .
Check the solution
$- 2 {(1)}^{2} + 4 (1) - 2 = 0$ .
Write the final solution
The root of $0 = - 2 x^{2} + 4 x - 2$ is $x = 1$ .

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Solving quadratic equations

Solve for $x$ : $(3 x + 2) (3 x - 4) = 0$
Solve for $x$ : $(5 x - 9) (x + 6) = 0$
Solve for $x$ : $(2 x + 3) (2 x - 3) = 0$
Solve for $x$ : $(2 x + 1) (2 x - 9) = 0$
Solve for $x$ : $(2 x - 3) (2 x - 3) = 0$
Solve for $x$ : $20 x + 25 x^{2} = 0$
Solve for $x$ : $4 x^{2} - 17 x - 77 = 0$
Solve for $x$ : $2 x^{2} - 5 x - 12 = 0$
Solve for $x$ : $- 75 x^{2} + 290 x - 240 = 0$
Solve for $x$ : $2 x = \frac{1}{3} x^{2} - 3 x + 14 \frac{2}{3}$
Solve for $x$ : $x^{2} - 4 x = - 4$
Solve for $x$ : $- x^{2} + 4 x - 6 = 4 x^{2} - 5 x + 3$
Solve for $x$ : $x^{2} = 3 x$
Solve for $x$ : $3 x^{2} + 10 x - 25 = 0$
Solve for $x$ : $x^{2} - x + 3$
Solve for $x$ : $x^{2} - 4 x + 4 = 0$
Solve for $x$ : $x^{2} - 6 x = 7$
Solve for $x$ : $14 x^{2} + 5 x = 6$
Solve for $x$ : $2 x^{2} - 2 x = 12$
Solve for $x$ : $3 x^{2} + 2 y - 6 = x^{2} - x + 2$

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

Aislinn Reply

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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

Jude

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what is viscosity?

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what is inorganic

emma

Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

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chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

Krampah Reply

2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

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answer

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progressive wave

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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

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Source: OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4

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