10TH ICSE Class MCQ on Quadratic Equations 2 ,10TH ICSE Class MCQ on Quadratic Equations 2 objective question ,10TH ICSE Class MCQ on Quadratic Equations 2 class 10th icse ,10TH ICSE Class MCQ on Quadratic Equations 2 Important questions.
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ICSE 10TH Class Objective Question On Quadratic Equations 2
Quadratic equations can be defined as a polynomial equation of a second degree, which implies that it comprises of a minimum of one term that is squared. The general form of the quadratic equation is:
ax² + bx + c = 0
where x is an unknown variable and a,b,c are numerical coefficients
Here, a ≠ 0 because if it equals zero then the equation will not remain quadratic anymore and it will become a linear equation, such as:
bx+c=0
Thus, this equation cannot be called a quadratic equation.
The terms a, b and c are also called quadratic coefficients.
The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of a polynomial lare the solutions for the given equation.
Quadratic Equation Definition
The polynomial equation whose highest degree is two is called a quadratic equation or sometimes just quadratics. It is expressed in the form of:
ax² + bx + c = 0
where x is the unknown variable and a, b and c are the constant terms.
Since the quadratic include only one unknown term or variable, thus it is called univariate. The power of variable x are always non-negative integers, hence the equation is a polynomial equation with highest power as 2.
The solution for this equation is the values of x, which are also called as zeros of the polynomial are the solution for which the equation is satisfied. In the case of quadratics, there are two roots or zeros of the equation. And if we put the values of roots or x in the Left-hand side of the equation, it will equal to zero. Therefore, they are called zeros.
Quadratics Formula
The formula for a quadratic equation is used to find the roots of the equation. Since quadratics have a degree equal to two, therefore there will be two solutions for the equation. Suppose, ax² + bx + c = 0 is the quadratic equation, then the formula to find the roots of this equation will be:
x = [-b±√(b2-4ac)]/2
The sign of plus/minus indicates there will be two solutions for x.
Examples of Quadratics
Beneath are the illustrations of quadratic equations of the form (ax² + bx + c = 0)
- x² –x – 9 = 0
- 5x² – 2x – 6 = 0
Examples of a quadratic equation with the absence of a ‘ C ‘- a constant term.
- -x² – 9x = 0
- x² + 2x = 0
Following are the examples of a quadratic equation in factored form
- (x – 6)(x + 1) = 0 [ result obtained after solving is x² – 5x – 6 = 0
- (x − 5)(x + 3) = 0 [result obtained after solving is x² − 2x − 15 = 0]
Below are the examples of a quadratic equation with an absence of linear co – efficient ‘ bx’
- 2x² – 64 = 0
- x² – 16 = 0
How to Solve Quadratic Equations?
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root
Factoring
- Begin with a equation of the form ax² + bx + c = 0
- Ensure that it is set to adequate zero.
- Factor the left-hand side of the equation by assuming zero on the right-hand side of the equation.
- Assign each factor equal to zero.
- Now solve the equation in order to determine the values of x.
Suppose if the main coefficient is not equal to one then deliberately, you have to follow a methodology in the arrangement of the factors.
Example:
2x²-x-6=0
(2x+3)(x-2)=0
2x+3=0
x=-3/2
x=2
Completing the Square
Let us learn this method with example.
Example: Solve 2x2 – x – 1 = 0.
First, move the constant term to the other side of the equation.
2x2 – x = 1
Dividing both sides by 2.
x2 – x/2 = ½
Add the square of half of the coefficient of x, (b/2a)2, on both the sides, i.e., 1/16
x2 – x/2 + 1/16 = ½ + 1/16
Now we can factor the right side,
(x-¼)2 = 9/16 = (¾)2
Taking root on both sides;
X – ¼ = ±3/4
Add ¼ on both sides
X = ¼ ± ¾
Therefore,
X = ¼ + ¾ = 4/4 = 1
X = ¼ – ¾ = -2/4 = -½
Using Quadratic Formula
For the given Quadratic equation of the form, ax² + bx + c = 0
Therefore the roots of the given equation can be found by:
x=(−b±√b2−4ac)/2a
where ± (one plus and one minus) represent two distinct roots of the given equation.
Taking the Square Root
We can use this method for the equations such as:
x2 – a2 = 0
Example: Solve x2 – 25 = 0.
x2 – 25 = 0
x2 = 25
Taking the roots both sides
√x2 = ±√25
x = ±√( 5 x 5)
x = ±5
10TH ICSE Class MCQ on Quadratic Equations 2
ANSWER KEY FOR QUADRATIC EQUATIONS 2
ANSWER KEY FOR PROBLEMS BASED ON QUADRATIC EQUATIONS
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10TH ICSE Class MCQ on Quadratic Equations 2