2.5.1 Quadratic Equations
Quadratic equations are polynomial equations of degree \( 2 \) in one variable of the form \( ax^2+bx+c=0 \)
Where \( a,b,c \in R \) and \( a≠0 \).
- It is the general form of a quadratic equation in which \( a \) is known as the leading coefficient and \( c \) is known as the absolute term
- There will always be two roots to the quadratic equation
- Roots can be real or imaginary
There are three basic methods for solving a quadratic equation
- Completing square
- Factorising
- Quadratic formula
2.5.1.1 Completing square
To solve a quadratic equation by completing square
- Take \( c \) on the other side of the equation
- Divide the equation with the coefficient of \( x^2 \) to make its coefficient \( 1 \)
- Half the coefficient of \( x \)
- Take its square and add it to both sides of equation
- Write it in the form \( (x+p)^2=q^2 \)
- Take square root on both sides to have roots
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Worked example:
Write \( x^2+10x+14 \) in the form \( (x+a)^2+b \).
Take \( c \) on the other side
\( x^2+10x=-14 \)
As coefficient of \( x^2 \) is \( 1 \) so we will proceed to the next topic
Half the coefficient of \( x \), take its square and add it to both sides
\( \frac{10}{2}=5 \)
\( x^2+10x+5^2=-14+5^2 \)
\( (x+5)^2=11 \)
Re-write in the form \( (x+a)^2+b \)
\( (x+5)^2+(-11) \)
Test yourself
Question 1:
- Write \( x^2+8x-9=0 \) in the form \( (x+k)^2+h \).
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- Use your answer to part 1. to solve the equation \( x^2+8x-9=0 \).
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Question 2:
\( x^2+4x-9=(x+a)^2+b \)
Find the value of \( a \) and \( b \).
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Question 3:
\( x^2-12x+a=(x+b)^2 \)
Find the value of \( a \) and \( b \)
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Question 4:
Write \( x^2-10x+14 \) in the form \( (x+a)^2+b \).
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2.5.1.2 Factorisation
To solve a quadratic equation by factorisation
- Write the equation in standard form, \( ax^2+bx+c=0 \)
- Factorize the expression
- Put each factor equal to zero
- Solve each equation
- Check your answer by plugging it into the original equation
Worked example:
Solve by factorisation \( 10r^2-23r+9=0\) .
We need two numbers that multiply to
\( 10×9=90 \) and add to \( -23 \)
\( -18 \) and \( -5 \) satisfy this
Rewrite the middle term using \( -18 \) and \( -5 \)
\( 10r^2-5r-18r+9=0 \)
Factorize the first two terms using \( 5r \) as the highest common factor and
the last two terms using \( 9 \) as the highest common factor
\( 5r(2r-1)-9(2r-1)=0 \)
\( (2r-1) \) is a common bracket, take it out
\( (2r-1)(5r-9)=0 \)
Put each factor equal to zero
\( (2r-1)=0, (5r-9)=0 \)
Solve each equation
\( r= \frac{1}{2},r= \frac{9}{5} \)
Test yourself
Question 1:
Solve \( 10m^2+9m-162=0 \).
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Question 2:
The \( n^{th} \) term of a sequence is \( 4n^2+n+3 \).
Find the value of \( n \) when the \( n^{th} \) term is \( 498 \).
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Question 3:
The difference between the areas of the two rectangles is \( 62cm^2 \).
Show that \( x^2+2x-63=0 \)
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Factorise \( x^2+2x-63=0 \)
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Solve the equation \( x^2+2x-63=0 \) to find the difference between the style=”font-size: calc(0.90375rem + 0.045vw);”>perimeters of the two rectangles.
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2.5.1.3 Quadratic formula
To solve a quadratic equation by using a quadratic formula
- Rewrite the equation in standard form \( ax^2+bx+c=0 \)
- Identify the values of \( a, b, c \)
- Write the quadratic formula
\( x=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \)
- Substitute in the values of \( a, b, c \)
- Simplify
- Round the answer
style=”font-size: calc(0.90375rem + 0.045vw);”>Discriminant:
The discriminant is the part of the formula that is under the square root \( (b^2-4ac) \)
- The sign of this value indicates the nature of solutions.
- If \( b^2-4ac>0 \) then there are two different solutions
- If \( b^2-4ac<0 \) then there is no real solution
- If \( b^2-4ac=0 \) then there are two repeated solutions
- Use your calculator to verify your final answer
Worked example:
Use the quadratic formula to solve \( 9x^2-12x-23=0 \)
Give your answer correct to \( 2 \) decimal places.
The equation is in standard form
Identify the values of \( a,b,c \)
\( a=9, b=-12, c=-23 \)
Quadratic formula is
\( x=\frac{-b \pm \sqrt {b^2-4ac}}{2a} \)
Substitute the values of \( a,b, c \)
\( x=\frac{-(-12)\sqrt{(-12)^2-4\times 9\times(-23)}}{2×9} \)
Simplify
\( x=\frac{12\pm\sqrt{972}}{18} \)
\( x=\frac{12+\sqrt{972}}{18} \) , \( x=\frac{12-\sqrt{972}}{18} \)
Use calculator to calculate the roots
\( x=2.398717… \), \( x=-1.065384… \)
Round to two decimal places
\( x=2.40 \), \( x=-1.07 \)
Test yourself
Question 1:
Solve the equation \( 3x^2-2x-10=0 \).
Show all your working and give your answer correct to two decimal places.
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Question 2:
Use the quadratic formula to solve the equation \( 3x^2+7x-11=0 \).
You must show all your working and give your answers correct to \( 2 \) decimal places.
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Question 3:
Solve the equation \( 6m^2+25m+16=0 \).
Show all your working and give your answers correct to \( 2 \) decimal places
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