Quadratic equations do not always appear in the standard form of ax + bx + c = 0.
Instead, several x terms, x terms, and numbers may appear on both sides of the equals sign. However, if all terms appear at least once, the equation can be rearranged in the standard form, and solved using the same methods as for simple equations.
x 2 + 11x + 13 = 2x − 7
x 2 + 9x + 20 = 0
Factors of +20 20, 1
The equation above is a typical quadratic equation, but cannot be solved by factorizing.
Listing all the possible factors and their sums in a table shows that there is no set of factors that add to b (3), and multiply to give c (1).
x 2 + 3x + 1 = 0
Factors of +1 1, 1
−1, −1
2
−2 Sum of factors
x + 5 = 0 x = –5 x + 4 = 0 x = –4
This equation is not written in standard quadratic form, but contains an x2 term and a term multiplied by x so it is known to be one. In order to solve it needs to be rearranged to equal 0.
Solve the quadratic equation by solving each of the bracketed expressions separately. Make each bracketed expression equal to 0, then find its solution. The two resulting values are the two solutions to the quadratic equation: –5 and –4.
Start by moving the numerical term from the right-hand side of the equals sign to the left by adding its opposite to both sides of the equation. In this case, –7 is moved by adding 7 to both sides.
It is now possible to solve the equation by factorizing. Draw a table for the possible numerical values of x. In one column, list all values that multiply together to give the c term, 20; in the other, add them together to see if they give the b term (9).
Write the correct pair of factors into parentheses and set them equal to 0. The two factors of the quadratic (x + 5) and (x + 4) multiply together to give 0, therefore one of the factors must be equal to 0.
Next, move the term multiplied by x to the left of the equals sign by adding its opposite to both sides of the equation. In this case, 2x is moved by subtracting 2x from both sides.
subtracting 2x from this side cancels out 2x
entire equation equals 0
7 has been added to this side, which cancels out –7, leaving 2x on its own 7 has been added to
this side (13 + 7 = 20)
these terms need to be moved to other side of equation for it to equal 0
a sum of +3 is needed as the b term is 3
stop when the factors add to the b term, 9
solve for first value
solve for second value
one possible solution is –5
another possible solution is –4
x 2 + 11x + 20 = 2x
both sets of numbers multiply together to give c (1)
adding –2x to 11x gives 9x
add the factors to find their sum
parentheses set next to each other are multiplied together
subtract 5 from both sides to isolate x
subtract 4 from both sides to isolate x all sets of numbers in this
column multiply to give 20
list possible factors of c = 20
b term (3) c term (1)
T h e qua dr atic fo rm ula QU ADR A TIC EQU A TIONS C AN BE SOL VED USING A FORMULA . T he quadr a tic f ormula
The quadratic formula can be used to solve any quadratic equation. Quadratic equations take the form ax² + bx + c = 0, where a, b, and c are numbers and x is the unknown.U sing the quadr a tic f ormula
To use the quadratic formula, substitute the values for a, b, and c in a given equation into the formula, then work through the formula to find the answers. Take great care with the signs (+, –) of a, b, and c.▷ A quadratic equation Quadratic equations include a number multiplied by x² , a number multplied by x, and a number by itself. ▷ The quadratic formula The quadratic formula allows any quadratic equation to be solved. Substitute the different values in the equation into the quadratic formula to solve the equation.
ax ²+ b x + c = 0 – 4 x ² + x – 3 = 8
x² x = –b b² – 4 a c x ² + 3 x – 2 = 0 x = – 3 3 ² – 4 × 1 × ( – 2 ) 2 × 1
+ +
– –
2a
Given a quadratic equation, work out the values of a, b, and c. Once these values are known, substitute them into the quadratic formula, making sure that their positive and negative signs do not change. In this example, a is 1, b is 3, and c is –2. substitut from equa into the f keeping their signs the same the values in the equation can be negative as well as positivequadratic equations are not always equal to 0 when an x appears without a number in front of it, x=1
number that multiplies xnumber that multiplies xnumber with no x terms this means add or subtract the value of b is 3the value of a is 1 the value of c is –2
SEE ALSO
177–179 Formulas
190–191 Factorizing quadratic equations Quadratic graphs 194 LOOK CLOSERQ uadra tic v aria tions
Quadratic equations are not always the same. They can include negative terms or terms with no numbers in front of them (“x” is the same as “1x”), and do not always equal 0.4.12 is the square root of 17 rounded to 2 decimal places. −3 + 4.12 = 1.12
−3 − 4.12 = −7.12
x = – 3 9 – ( – 8 ) x = – 3 1 7 x = – 3 4 .1 2 x = – 3 + 4 .1 2 x = 1 .1 2 x = 0. 5 6
x = – 3 – 4 .1 x = – 7. 12 x = –3 .5 6
2 2 2 2 2
2 2
+ + + – – –
Work through the formula step- by-step to find the answer to the equation. First simplify the values under the square root sign. Work out the square of 3 (which equals 9), then work out the value of 4 × 1 × –2 (which equals –8). Once the sum is simplified, it must be split to find the two answers— one when the second value is subtracted from the first, and the other where they are added. Subtract the second value on the top part of the fraction from the first value; here the values are –3 and –4.12.
Add the two values on the top part of the fraction; here the values are –3 and 4.12. Divide the top part of the fraction by the bottom part to find an answer.
Divide the top part of the fraction by the bottom part to find an answer.
Work out the numbers under the square root sign: 9–(–8) equals 9 + 8, which equals 17. Then, use a calculator to find the square root of 17. Give both answers, because quadratic equations always have two solutions.
two minus signs cancel out, so 9–(–8) = 9 + 8 quadratic equations always have two solutions
3 × 3 = 94 × 1 × (–2) = –8 9 + 8 = 17
+ –
add values in each purple row to find y values