The polynomial which can be expressed in the form of ax2 + bx + c = 0, then we say that the equation is in the form of quadratic polynomial. Here we say that a, b, c are the real Numbers and we must remember that a <> 0, since if we have a = 0, the equation will convert into a linear equation in place of the quadratic equation. If we say that alpha (α) and beta (β) are the two roots of the quadratic equation and their sum of the root i.e. α + β is written as = - coefficient of ‘x’ / coefficient of x2.
Define Quadratic EquationBack to Top
Suppose we have an equation ax2 + bx + c = 0, so first of all we will find roots of this equation, so by using quadratic formula,
x = -b ∓ (√(b2 – 4ac))/2a , are two solutions of this quadratic equation.
x = -b + (√(b2 – 4ac))/2a , x = -b -(√(b2 – 4ac))/2a .
Here (b2 – 4ac) is called as discriminant, it is represented by Δ.
Case 1: If Δ > 0 , (discriminant is positive) then in this case there are two roots and both are Real Numbers.
Roots = (-b +(√ Δ ))/2a , (-b - (√Δ))/2a .
Case 2: If Δ = 0, in this case we have only one real root (double root).
Root = - b / 2a .
Case 3 : If Δ < 0 (discriminant is negative), in this case there are two roots, they both are non real.
Root = -b / 2a + ί( √-Δ ))/2a , -b / 2a - ί(√-Δ))/2a.
Quadratic Equations by FactoringBack to Top
Quadratic FormulaBack to Top
⇨ x = - b + √ (b2 – 4ac) / 2a, its alternate form also given by:
⇨ x = 2c / -b +√ (b2 – 4ac).
Now we will see how to find factors of quadratic equation using Quadratic Formula. To find factors we need to follow some steps:
Step 1: To find factor first we take a quadratic equation. Let quadratic equation is 5p2 – p + 4 = 0.
Step 2: Use above formula to find factors of expression. We know that quadratic formula is given by: ⇨ x = - b + √ (b2 – 4ac) / 2a. In equation the value of ‘a’ is 5, value of ‘b’ is -1 and value of ‘c’ is 4.
Step 3: Put these values in formula. On putting these values we get:
⇨ x = - b + √ (b2 – 4ac) / 2a,
⇨ x = - (-1) + √ ((-1)2 – 4 * 5 * 4) / 2 (5);
⇨ x = -1 + √ (-1 – 80) / 10;
⇨ x = -1 + √ 81 / 10; so here we will get two factors of given equation.
X = -1 + √ 81 / 10 and X = -1 - √ 81 / 10.
This is how we find factors of quadratic equation using quadratic formula.
Square Root PropertyBack to Top
We can solve a Quadratic Equation using square root property where we can take square root of a squared expression to eliminate exponent in the expression. This property has to be applied to both sides of equation to keep things even-handed. If any other arithmetic operations are needed to be performed, go for them to find answer from there. Note that at last you will be getting two solutions for 'a'.
Square root property can be used to solve equations like (x - 10)² = 4 sort. Begin solving it by taking square root on both sides to eliminate exponent part: we get x - 10 = √4. Place plus or minus symbol in front of square root of "4" to symbolize that two solutions are possible and so two values of x: x - 10 = ± √4. Perform the operation, retaining the plus or minus symbol: x - 10 = ± 2.
Now we are left with two equations given as x – 10 = 2 and x – 10 = - 2. Solve them for 'x'. We get, x = 12 or x = 8. This way we can solve even more complex equations using square root property.
Completing the SquareBack to Top
(a + b) 2 = a2 + b2 + 2ab and
(a - b) 2 = a2 + b2 - 2ab,
When you have a Quadratic Equation of form ax² + bx + c which is not possible to be factorized, you can use technique called completing the square. To complete the square means creating a polynomial with three terms that results into a perfect square.
To start with, rewrite the quadratic expression given as ax² + bx + c and move the constant term 'c' to right side of equation to get the form ax² + bx = - c. Divide this complete equation constant factor “a” if a≠ 1 to get x² + (b / a) x = -c / a.
Divide coefficient of 'x' i.e. (b / a) by 2 and it now becomes (b / 2a) and then square it to get (b / 2a) ². Add (b / 2a)² to both sides of equation to get:
x² + (b / a) x + (b / 2a)² = -c / a + (b / 2a) ².
Next step would we writing left side of equation as perfect square:
[x + (b / 2a)] ² = -c / a + (b / 2a) ².
For example, let’s take an expression: 4x² + 16x - 20. Where, a = 4, b = 16 and c = -20.
Moving constant 'c' to right side we get 4x² + 16x = 20. Next divide both sides of equation by 4 to get: x² + 4x = 20 / 4. Taking half of 4 which is coefficient of 'x' and then squaring it to get:
(4 / 2) ² = 4,
Add 4 to your equation:
x² + 4x + 4 = 5 + 4,
or x² + 4x + 4 = 9,
Making left side a perfect square we get:
(x + 2) 2 = 9,
or x = 1, -5.