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You đang tìm kiếm từ khóa What is the quadratic equation if the sum of roots is 2 and product of roots is 5? được Cập Nhật vào lúc : 2022-11-26 09:50:14 . Với phương châm chia sẻ Bí kíp Hướng dẫn trong nội dung bài viết một cách Chi Tiết 2022. Nếu sau khi đọc Post vẫn ko hiểu thì hoàn toàn có thể lại Comment ở cuối bài để Ad lý giải và hướng dẫn lại nha.In this article we cover quadratic equations – definitions, formats, solved problems and sample questions for practice.
Nội dung chính Show- Solved examples of Quadratic equations
- Discriminant
- Quadratic Equations Quiz: Solve the following
- Which of the following equation has the product of two roots as 5?
- What will be the quadratic equation if a 2 ß 5?
- What is the formula in the sum of the roots of quadratic equation?
A quadratic equation is a polynomial whose highest power is the square of a variable (x2, y2 etc.)
DefinitionsA monomial is an algebraic expression with only one term in it.
Example: x3, 2x, y2, 3xyz etc.
A polynomial is an algebraic expression with more than one term in it.
Alternatively it can be stated as –
A polynomial is formed by adding/subtracting multiple monomials.
Example: x3+2y2+6x+10, 3x2+2x-1, 7y-2 etc.
A polynomial that contains two terms is called a binomial expression.
A polynomial that contains three terms is called a trinomial expression.
A standard quadratic equation looks like this:
ax2+bx+c = 0
Where a, b, c are numbers and a≥1.
a, b are called the coefficients of x2 and x respectively and c is called the constant.
The following are examples of some quadratic equations:
1) x2+5x+6 = 0 where a=1, b=5 and c=6.
2) x2+2x-3 = 0 where a=1, b=2 and c= -3
3) 3x2+2x = 1
→ 3x2+2x-1 = 0 where a=3, b=2 and c= -1
4) 9x2 = 4
→ 9x2-4 = 0 where a=9, b=0 and c= -4
For every quadratic equation, there can be one or more than one solution. These are called the roots of the quadratic equation.
For a quadratic equation ax2+bx+c = 0,
the sum of its roots = –b/a and the product of its roots = c/a.
A quadratic equation may be expressed as a product of two binomials.
For example, consider the following equation
x2-(a+b)x+ab = 0
x2-ax-bx+ab = 0
x(x-a)-b(x-a) = 0
(x-a)(x-b) = 0
x-a = 0 or x-b = 0
x = a or x=b
Here, a and b are called the roots of the given quadratic equation.
Now, let’s calculate the roots of an equation x2+5x+6 = 0.
We have to take two numbers adding which we get 5 and multiplying which we get 6. They are 2 and 3.
Let us express the middle term as an addition of 2x and 3x.
→ x2+2x+3x+6 = 0
→ x(x+2)+3(x+2) = 0
→ (x+2)(x+3) = 0
→ x+2 = 0 or x+3 = 0
→ x = -2 or x = -3
This method is called factoring.
We saw earlier that the sum of the roots is –b/a and the product of the roots is c/a. Let us verify that.
Sum of the roots for the equation x2+5x+6 = 0 is -5 and the product of the roots is 6.
The roots of this equation -2 and -3 when added give -5 and when multiplied give 6.
Solved examples of Quadratic equations
Let us solve some more examples using this method.
Problem 1: Solve for x: x2-3x-10 = 0
Solution:
Let us express -3x as a sum of -5x and +2x.
→ x2-5x+2x-10 = 0
→ x(x-5)+2(x-5) = 0
→ (x-5)(x+2) = 0
→ x-5 = 0 or x+2 = 0
→ x = 5 or x = -2
Problem 2: Solve for x: x2-18x+45 = 0
Solution:
The numbers which add up to -18 and give +45 when multiplied are -15 and -3.
Rewriting the equation,
→ x2-15x-3x+45 = 0
→ x(x-15)-3(x-15) = 0
→ (x-15) (x-3) = 0
→ x-15 = 0 or x-3 = 0
→ x = 15 or x = 3
Till now, the coefficient of x2 was 1. Let us see how to solve the equations where the coefficient of x2 is greater than 1.
Problem 3: Solve for x: 3x2+2x =1
Solution:
Rewriting our equation, we get 3x2+2x-1= 0
Here, the coefficient of x2 is 3. In these cases, we multiply the constant c with the coefficient of x2. Therefore, the product of the numbers we choose should be equal to -3 (-1*3).
Expressing 2x as a sum of +3x and –x
→ 3x2+3x-x-1 = 0
→ 3x(x+1)-1(x+1) = 0
→ (3x-1)(x+1) = 0
→ 3x-1 = 0 or x+1 = 0
→ x = 1/3 or x = -1
Problem 4: Solve for x: 11x2+18x+7 = 0
Solution:
In this case, the sum of the numbers we choose should equal to 18 and the product of the numbers should equal 11*7 = 77.
This can be done by expressing 18x as the sum of 11x and 7x.
→ 11x2+11x+7x+7 = 0
→ 11x(x+1) +7(x+1) = 0
→ (x+1)(11x+7) = 0
→ x+1 = 0 or 11x+7 = 0
→ x = -1 or x = -7/11.
The factoring method is an easy way of finding the roots. But this method can be applied only to equations that can be factored.
For example, consider the equation x2+2x-6=0.
If we take +3 and -2, multiplying them gives -6 but adding them doesn’t give +2. Hence this quadratic equation cannot be factored.
For this kind of equations, we apply the quadratic formula to find the roots.
The quadratic formula to find the roots,
x = [-b ± √(b2-4ac)] / 2a
Now, let us find the roots of the equation above.
x2+2x-6 = 0
Here, a = 1, b=2 and c= -6.
Substituting these values in the formula,
x = [-2 ± √(4 – (4*1*-6))] / 2*1
→ x = [-2 ± √(4+24)] / 2
→ x = [-2 ± √28] / 2
When we get a non-perfect square in a square root, we usually try to express it as a product of two numbers in which one is a perfect square. This is for simplification purpose. Here 28 can be expressed as a product of 4 and 7.
→ x = [-2 ± √(4*7)] / 2
→ x = [-2 ± 2√7] / 2
→ x = 2[ -1 ± √7] / 2
→ x = -1 ± √7
Hence, √7-1 and -√7-1 are the roots of this equation.
Let us consider another example.
Solve for x: x2 = 24 – 10x
Solution:
Rewriting the equation into the standard quadratic form,
x2 +10x-24 = 0
What are the two numbers which when added give +10 and when multiplied give -24? 12 and -2.
So this can be solved by the factoring method. But let’s solve it using the new method, applying the quadratic formula.
Here, a = 1, b = 10 and c = -24.
x = [-10 ± √(100 – 4*1*-24)] / 2*1
x = [-10 ± √(100-(-96))] / 2
x = [-10 ± √196] / 2
x = [-10 ± 14] / 2
x = 2 or x= -12 are the roots.
Discriminant
For an equation ax2+bx+c = 0, b2-4ac is called the discriminant and helps in determining the nature of the roots of a quadratic equation.
If b2-4ac > 0, the roots are real and distinct.
If b2-4ac = 0, the roots are real and equal.
If b2-4ac < 0, the roots are not real (they are complex).
Consider the following example:
Problem: Find the nature of roots for the equation x2+x+12 = 0.
Solution:
b2-4ac = -47 for this equation. So it has complex roots. Let us verify this.
→ [ -1±√(1-48)] / 2(1)
→ [-1±√-47] / 2
√-47 is usually written as i √47 indicating it’s an imaginary number.
Hence verified.
Quadratic Equations Quiz: Solve the following
Problem 1: Click here
Answer 1: Click here
Problem 2: Click here
Answer 2: Click here
Problem 3: Click here
Answer 3: Click here
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Quadratic Equations; Your Complete Guide
Which of the following equation has the product of two roots as 5?
Answer : `x^(2)-6x+5=0`.What will be the quadratic equation if a 2 ß 5?
What will be the quadratic equation if α=2,β=5 It can be written as x2−(□+□)x+□×□=0.What is the formula in the sum of the roots of quadratic equation?
Thus we can express the sum of the roots in terms of the coefficients a,b,c of the quadratic as α+β=−ba. In the case when the quadratic does not cross the x-axis, the corresponding quadratic equation ax2+bx+c=0 has no real roots, but it will have complex roots (involving the square root of negative numbers). Tải thêm tài liệu liên quan đến nội dung bài viết What is the quadratic equation if the sum of roots is 2 and product of roots is 5? Patterns of polynomials
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