Question

How do you write $y = 4{(x - 3)^2} + 1$ in standard form?

Answer 1

Hint: In this question, we want to convert the given expression into the standard form. To solve the question, we apply the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$. Then simplify the expression. The answer will be in the form of the quadratic equation. The general form of the quadratic equation is $a{x^2} + bx + c = 0$. Where ‘a’ is the coefficient of ${x^2}$, ‘b’ is the coefficient of x and ‘c’ is the constant term.

Complete step by step answer:
In this question, we want to convert the given expression in its standard form.
The expression is:
$ \Rightarrow y = 4{(x - 3)^2} + 1$
To solve this expression, let us apply the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$.
Here, the value of ‘a’ is ‘x’ and the value of ‘b’ is 3.
Put the value of ‘a’ and ‘b’ in the algebraic identity.
$ \Rightarrow y = 4\left[ {{{\left( x \right)}^2} - 2\left( x \right)\left( 3 \right) + {{\left( 3 \right)}^2}} \right] + 1$
Let us simplify the above expression. The square of x is ${x^2}$, the square of 3 is 9.
That is equal to,
$ \Rightarrow y = 4\left( {{x^2} - 6x + 9} \right) + 1$
Let us apply multiplication. Multiply 4 with the bracket to remove the bracket.
Therefore,
$ \Rightarrow y = 4{x^2} - 24x + 36 + 1$
Now, let us apply the addition on the right-hand side.
The addition of 36 and 1 is 37.
Therefore,
$ \Rightarrow y = 4{x^2} - 24x + 37$
Hence, the simple form of the given expression is $y = 4{x^2} - 24x + 37$.

Note: he quadratic equation is $a{x^2} + bx + c = 0$. Where ‘a’ is the coefficient of ${x^2}$, ‘b’ is the coefficient of x and ‘c’ is the constant term. Here, ‘a’, ‘b’, and ‘c’ are known values, ‘x’ is the variable or unknown. And ‘a’ can’t be 0. The solution of the quadratic equation is where it is equal to zero. They are also called roots or zeros.
Here is a list of methods to solve quadratic equations:
1. Factorization
2. Completing the square
3. Using graph
4. Quadratic formula