Quadratic equation Flashcards
What are quadratic equations?
Are equations that contains a squared variable, such as x², and no higher power.
How many solutions has a quadratic equation?
Usually a quadratic equation has two solutions.
How to factor a quadratic equation?
Rewrite the expression as a product of two sums:
x² + 5x + 6 = (x + 2)(x + 3)
- in that case, note that the numbers should be multiple of 6 and its sum should be equal to 5. You can try to find the factor pairs of the final number (the constant) to identify the factor pairs that will result in the coefficient (5) when added.
How to factor 3x² + 21x + 36?
Pull out the common factor first:
3(x² + 7x + 12)
3(x + 3)(x + 4)
How to factor a quadratic equation with a negative x², e.g., -x² + 9x - 18?
Factor out a common factor of -1 first. That will flip the sign on every term:
-(x² - 9x + 18) = - (x - 3)(x - 6)
How to factor 2z² - z - 15?
If you cannot pull out a common factor from the x² term without turning coefficients into fractions, then keep a coefficient on one or even both x’s in your factored form.
- Find the factor pairs that are likely to complete the factorization and try them by chance until finding the correct combination
(2z + 5) (z - 3) = 2z² - 6z + 5z - 15 = 2z² - z -15
How to solve a quadratic equation with a non-squared variable (z² + z - 8 = 4)?
1) rearrange the equation to make one side equal 0. The other side will contain a quadratic expression.
2) Factor the quadratic expression. The equation will look like this:
(something)(something else) = 0
3) set each factor to equal to 0.
Something = 0 or Something else = 0
z² + z - 8 = 4
z² + z - 12 = 0
(z + 4)(z - 3) = 0
z = -4 or z = 3
How to solve a quadratic with no non-squared variable (x² = 9)?
Take positive and negative square roots.
x² = 9
x = √9
x = 3 or -3
How to solve (y + 1)² = 16?
Take positive and negative square roots.
(y + 1)² = 16
y + 1 = √16
y + 1 = 4 or -4
y = 3 or -5
How to solve equations to higher powers?
x³ = 3x² - 2x
First set one side of the equation equal to 0
x³ = 3x² - 2x
x³ - (3x² - 2x) = 0
x³ - 3x² + 2x = 0
As the equation has all terms with an “x”, you can factor “x” as a common factor
x(x² - 3x + 2) = 0
Now rewrite the equation:
x(x - 2)(x - 1) = 0
Set each factors on the left side as equal to zero:
x = 0
x - 2 = 0
x = 2
x - 1 = 0
x = 1
How to solve:
x² - 2x - 3 / x + 1
Factor the expressions and cancel the like terms.
x² - 2x - 3 / x + 1
= (x - 3)(x + 1) / x + 1 (cancel the denominator with the numerator)
= x - 3
How to solve y - x / x - y
Rewrite the numerator. Pull a -1 out of the (x - y) term:
y - x / x - y
= - (x - y) / x - y (cancel the denominator with the numerator)
= -1
