Question

zeroes of the polynomial ax² - 5x+ c such that α + β = α β = 10, then the values of a and c are ------------

Answer 1

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Answer 2

Answer:

The value of a is $\bold{\sf\dfrac{1}{2}}$.

The value of c is 5.

Step-by-step-explanation:

The given quadratic equation is

$\sf\:ax^2\:-\:5x\:+\:c\:=\:0$

We have given that,

$\sf\:\alpha\:+\:\beta\:=\:\alpha\:.\:\beta\:=\:10$

We know that,

$\pink{\sf\:\alpha\:+\:\beta\:=\:-\:\frac{b}{a}}\\\\\\\implies\sf\:10\:=\:-\:\frac{-\:5}{a}\\\\\\\implies\sf\:10\:=\:\frac{5}{a}\\\\\\\implies\sf\:a\:=\:\cancel{\frac{5}{10}}\\\\\\\implies\boxed{\red{\sf\:a\:=\:\dfrac{1}{2}}}$

Now, we know that,

$\pink{\sf\:\alpha\:.\:\beta\:=\:\dfrac{c}{a}}\\\\\\\implies\sf\:10\:=\:\frac{c}{a}\\\\\\\implies\sf\:10\:=\:\dfrac{c}{\dfrac{1}{2}}\\\\\\\implies\sf\:c\:\times\:2\:=\:10\\\\\\\implies\sf\:c\:=\:\cancel{\frac{10}{2}}\\\\\\\implies\boxed{\red{\sf\:c\:=\:5}}$

Additional Information:

1. Quadratic Equation:

An equation having a degree '2' is called quadratic equation.

The general form of quadratic equation is

ax² + bx + c = 0

Where, a, b, c are real numbers and a ≠ 0.

2. Roots of Quadratic Equation:

The roots means nothing but the value of the variable given in the equation.

3. Methods of solving quadratic equation:

There are mainly three methods to solve or find the roots of the quadratic equation.

A) Factorization method

B) Completing square method

C) Formula method

4. Formula to solve quadratic equation:

$\boxed{\red{\sf\:x\:=\:\dfrac{-\:b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}}$