Question

Determine a quadratic equation with the given roots
1. 2 and -1
2. -2 and -6
3. 3 and 4
​

Answer 1

Answer:

$if \: \alpha \: and \: \beta \: are \: roots \: of \: quadatic \: eq. \: then \: \\ eq. \: is \: \\ {x}^{2} - ( \alpha + \beta )x + \alpha \beta = 0$

$1. \: \alpha + \beta = 2 - 1 = 1 \: \: and \: \: \alpha \beta = - 1 \times 2 = - 2 \\ so \: eq \: is \\ {x}^{2} - x - 2 = 0$

solve other same as above.

❤Thanks

Answer 2

Answer :

• The quadratic equation formed by the roots 2 and (-1) is x² - x - 2 = 0

• The quadratic equation formed by the roots (-2) and (-6) is x² + 8x + 12 = 0

• The quadratic equation formed by the roots 3 and 4 is x² - 7x + 12 = 0

Explanation :

Given :

• Roots of the first quadratic equation = 2 and (-1).

• Roots of the second quadratic equation = (-2) and (-6)

• Roots of the second quadratic equation formed = 3 and 4

To find :

• Quadratic equations whose roots are given.

Knowledge required :

Formula for forming a quadratic equation :

⠀⠀⠀⠀⠀⠀⠀⠀⠀x² - (α + β)x + αβ = 0

Where :

• α and β are the two roots of the equation

Solution :

Quadratic equation with roots of 2 and (-1) :

By using the formula for forming a quadratic equation and substituting the values in it, we get :

==> x² - (α + β)x + αβ = 0

==> x² - (2 + (-1))x + 2 × (-1) = 0

==> x² -(2 - 1)x + (-2) = 0

==> x² - x - 2 = 0

∴ Quadratic Equation = x² - x - 2 = 0

Quadratic equation with roots of (-2) and (-6) :

By using the formula for forming a quadratic equation and substituting the values in it, we get :

==> x² - (α + β)x + αβ = 0

==> x² - [(-2) + (-6)]x + (-2) × (-6) = 0

==> x² - (-2 - 6)x + 12 = 0

==> x² - (-8)x + 12 = 0

==> x² + 8x + 12 = 0

∴ Quadratic Equation = x² + 8x + 12 = 0

Quadratic equation with roots of 3 and 4 :

By using the formula for forming a quadratic equation and substituting the values in it, we get :

==> x² - (α + β)x + αβ = 0

==> x² - (3 + 4)x + 3 × 4 = 0

==> x² - 7x + 12 = 0

∴ Quadratic Equation = x² - 7x + 12 = 0

Therefore,

• The quadratic equation formed by the roots 2 and (-1) is x² - x - 2 = 0

• The quadratic equation formed by the roots (-2) and (-6) is x² + 8x + 12 = 0

• The quadratic equation formed by the roots 3 and 4 is x² - 7x + 12 = 0