Question

Find the quadratic equation with roots $ 4 $ and $ 5 $

Answer 1

Hint: First, assume that a quadratic equation in any form.
Quadratic means at most power two terms, applied by the condition that the sum of the given roots and product of the given roots.
After finding the values of the sum of the two terms and the product of the two terms, these values can be substituted back into the general equation.
Formula used: The general quadratic equation formula is $ a{x^2} - bx + c = 0 $
The sum of the given roots can be expressed as $ \dfrac{{ - b}}{a} $ and the product is $ \dfrac{c}{a} $

Complete step-by-step answer:
From the given that the roots of the quadratic equations are $ 4 $ and $ 5 $ .
Since the general quadratic equation can be written as in the form of $ a{x^2} - bx + c = 0 $ where a is the quadratic root with at most two-degree it will be not represented as any terms, b is the sum of the given terms.
Divide the quadratic form by $ a $ then we get $ {x^2} - \dfrac{b}{a}x + \dfrac{c}{a} = 0 $
In quadratic we only have at most two values and hence the sum of the terms for only two values is represented as $ \dfrac{{ - b}}{a} $ .
And for $ \dfrac{c}{a} $ is the product of the given quadratic equation
Hence these terms are represented as $ \dfrac{b}{a} = {1^{st}} + {2^{nd}} $ (sum of the two given root terms one and second)
And c is representing as the product, hence $ \dfrac{c}{a} = {1^{st}} \times {2^{nd}} $ (product of the two given terms, one and second)
The given roots are $ 4 $ and $ 5 $ .
Apply this in above b and c we get, $ \dfrac{b}{a} = {1^{st}} + {2^{nd}} \Rightarrow 4 + 5 $ thus by the addition operation we get, $ \dfrac{{ - b}}{a} = 9 $
Similarly for the c, we get $ \dfrac{c}{a} = {1^{st}} \times {2^{nd}} \Rightarrow 4 \times 5 = 20 $ (by the multiplication operation)
Hence, we get the $ \dfrac{b}{a} = 9 \Rightarrow \dfrac{{ - b}}{a} = - 9 $ and $ \dfrac{c}{a} = 20 $
Therefore, the quadratic equation with roots $ 4 $ and $ 5 $ are $ {x^2} - 9x + 20 = 0 $

Note: Be careful with the value b, because in the given general formula it is in the negative sign, so if we apply the terms as $ {x^2} + 9x + 20 = 0 $ then the answer will be incorrect.
The addition is the sum of adding two or more than two numbers are variables.
We will see what is multiplication, Multiplicand refers to the number multiplied. The multiplier is the number that refers to the number which multiplies the first number.
If the value $ a = 0 $ then the function is changed as a linear function because $ a{x^2} - bx + c = 0 \Rightarrow bx + c = 0 $ . So that value of a will never be zero.