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Mark
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Maths
Solving Quadratics by Factorising
How do I solve a quadratic equation using factorisation?
- Rearrange it into the form ax2 + bx + c = 0
- zero must be on one side
- it is easier to use the side where a is positive
- zero must be on one side
- Factorise the quadratic and solve each bracket equal to zero
- If (x + 4)(x - 1) = 0, then either x + 4 = 0 or x - 1 = 0
- Because if A × B = 0, then either A = 0 or B = 0
- If (x + 4)(x - 1) = 0, then either x + 4 = 0 or x - 1 = 0
- Factorise the quadratic and solve each bracket equal to zero
- To solve
- …solve “first bracket = 0”:
- x – 3 = 0
- add 3 to both sides: x = 3
- …and solve “second bracket = 0”
- x + 7 = 0
- subtract 7 from both sides: x = -7
- The two solutions are x = 3 or x = -7
- The solutions have the opposite signs to the numbers in the brackets
- …solve “first bracket = 0”:
- To solve
- …solve “first bracket = 0”
- 2x – 3 = 0
- add 3 to both sides: 2x = 3
- divide both sides by 2: x =
- …solve “second bracket = 0”
- 3x + 5 = 0
- subtract 5 from both sides: 3x = -5
- divide both sides by 3: x =
- The two solutions are x = or x =
- …solve “first bracket = 0”
- To solve
- it may help to think of x as (x – 0) or (x)
- …solve “first bracket = 0”
- (x) = 0, so x = 0
- …solve “second bracket = 0”
- x – 4 = 0
- add 4 to both sides: x = 4
- The two solutions are x = 0 or x = 4
- It is a common mistake to divide both sides by x at the beginning -you will lose a solution (the x = 0 solution)
Exam Tip
- Use a calculator to check your final solutions!
- Calculators also help you to factorise (if you're struggling with that step)
- A calculator gives solutions to as x = and x =
- "Reverse" the method above to factorise!
- Warning: a calculator gives solutions to 12x2 + 2x – 4 = 0 as x = and x =
- But 12x2 + 2x – 4 ≠ as these brackets expand to 6x2 + ... not 12x2 + ...
- Multiply by 2 to correct this
- 12x2 + 2x – 4 =
Worked example
(a)
Solve
Set the first bracket equal to zero
x – 2 = 0
Add 2 to both sides
x = 2
Set the second bracket equal to zero
x + 5 = 0
Subtract 5 from both sides
x = -5
Write both solutions together using “or”
x = 2 or x = -5
(b)
Solve
Set the first bracket equal to zero
8x + 7 = 0
Subtract 7 from both sides
8x = -7
Divide both sides by 8
x =
Set the second bracket equal to zero
2x - 3 = 0
Add 3 to both sides
2x = 3
Divide both sides by 2
x =
Write both solutions together using “or”
x = or x =
(c)
Solve
Do not divide both sides by x(this will lose a solution at the end)
Set the first “bracket” equal to zero
(x) = 0
Solve this equation to find x
x = 0
Set the second bracket equal to zero
5x - 1 = 0
Add 1 to both sides
5x = 1
Divide both sides by 5
x =
Write both solutions together using “or”
x = 0 or x =
Solving by Completing the Square
How do I solve a quadratic equation by completing the square?
- To solvex2 + bx + c = 0
- replace the first two terms, x2 + bx, with (x + p)2 - p2 where p is half of b
- this is called completing the square
- x2 + bx + c = 0 becomes
- (x + p)2 - p2 + c = 0 where p is half of b
- x2 + bx + c = 0 becomes
- rearrange this equation to make x the subject (using ±√)
- For example, solve x2 + 10x + 9 = 0 by completing the square
- x2 + 10x becomes(x + 5)2 - 52
- so x2 + 10x + 9 = 0 becomes(x + 5)2 - 52 + 9 = 0
- make x the subject (using ±√)
- (x + 5)2 - 25+ 9 = 0
- (x + 5)2 = 16
- x + 5 = ±√16
- x = ±4 - 5
- x = -1 orx = -9
- If the equation is ax2 + bx + c = 0 with a number in front of x2, then divide both sides by a first, before completing the square
How does completing the square link to the quadratic formula?
- The quadratic formula actually comes from completing the square to solve ax2 + bx + c = 0
- a, b and c are left as letters, to be as general as possible
- You can see hints of this when you solve quadratics
- For example, solving x2 + 10x + 9 = 0
- by completing the square,(x + 5)2 = 16 sox = ± 4 - 5 (from above)
- by the quadratic formula, = ± 4 - 5 (the same structure)
- For example, solving x2 + 10x + 9 = 0
Exam Tip
- When making x the subject to find the solutions at the end, don't expand the squared brackets back out again!
- Remember to use ±√ to get two solutions
Worked example
Solve by completing the square
Divide both sides by 2 to make the quadratic start with x2
Halve the middle number, -4, to get -2
Replace the first two terms, x2 - 4x, with (x - 2)2 - (-2)2
Simplify the numbers
Add 16 to both sides
Square root both sides
Include the ± sign to get two solutions
Add 2 to both sides
Work out each solution separately
x = 6 or x = -2
Quadratic Formula
How do I use the quadratic formula to solve a quadratic equation?
- A quadratic equation has the form:
ax2 + bx + c = 0 (as long as a ≠ 0)
- you need "= 0" on one side
- The quadratic formula is a formula that gives both solutions:
- Read off the values of a, b and c from the equation
- Substitute theseinto the formula
- write this line of working in the exam
- Put brackets around any negative numbers being substituted in
- To solve 2x2 - 7x - 3 = 0 using the quadratic formula:
- a = 2, b = -7 and c = -3
- Type this into a calculator
- once with + for ± and once with - for ±
- The solutions are x = 3.886 and x = -0.386 (to 3 dp)
- Rounding is often asked for in the question
- The calculator also gives these solutions in exact form (surd form), if required
- x = and x =
What is the discriminant?
- The part of the formula under the square root (b2 – 4ac) is called the discriminant
- The sign of this value tells you if there are 0, 1 or 2 solutions
- If b2 – 4ac > 0 (positive)
- then there are 2 different solutions
- If b2 – 4ac = 0
- then there is only 1 solution
- sometimes called "two repeated solutions"
- If b2 – 4ac < 0 (negative)
- then there are no solutions
- If your calculator gives you solutions with i terms in, these are "complex" and not what we are looking for
- Interestingly, if b2 – 4ac is a perfect square number ( 1, 4, 9, 16, …) then the quadratic expression could have been factorised!
- If b2 – 4ac > 0 (positive)
Can I use my calculator to solve quadratic equations?
- Yes tocheck your final answers, but a method must still be shown as above
Exam Tip
- Make sure the quadratic equation has "= 0" on the right-hand side, otherwise it needs rearranging first
- Always look for how the question wants you to leave your final answers
- for example, correct to 2 decimal places
Worked example
Use the quadratic formula to find the solutions of the equation 3x2 - 2x - 4 = 0, giving your answers correct to 3 significant figures.
Write down the values of a, b and c
a = 3, b = -2, c = -4
Substitute these values into the quadratic formula,
Put brackets around any negative numbers
Input this into a calculator
Use + for ± to get the first solution
x = 1.53518...
Input this into a calculator a second time
Use - for ± to get the second solution
x = 0.86851...
Present both answers together (using the word "or" between them)
Round the answers correct to 3 significant figures (note how this affects the number of decimal places)
x = 1.54 or x = 0.869
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