Question

How do you solve ${x^2} + 4x + 4 = 7$ and simplify the answer in simplest radical form?

Answer 1

Hint:
In this question we have to solve the given polynomial. Now take all the terms to one side and then the given polynomial will be in the form of the polynomial \[a{x^2} + bx + c\], where "\[a\]", "\[b\]", and “\[c\]" are real numbers and the Quadratic Formula is derived from the process of completing the square, and is formally stated as:
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], now substituting the values in the formula we will get the required value.

Complete step by step solution:
Quadratic equations are the equations that are often called second degree. It means that it consists at least one term which is squared, the general form of quadratic equation is \[a{x^2} + bx + c = 0\], where "\[a\]", "\[b\]", and "\[c\]"are numerical coefficients or constant, and the value of \[x\] is unknown. And one fundamental rule is that the value of, the first constant cannot be zero in a quadratic equation.
Now the given quadratic equation is,
${x^2} + 4x + 4 = 7$,
Now subtract 7 on both sides of the equation, we get,
$ \Rightarrow {x^2} + 4x + 4 - 7 = 7 - 7$,
Now simplifying the equation we get,
$ \Rightarrow {x^2} + 4x - 3 = 0$,
Now using the quadratic formula, which is given by\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\],
Here \[a = 1\], \[b = 4\], \[c = - 3\],
Now substituting the values in the formula we get,
\[ \Rightarrow x = \dfrac{{ - \left( 4 \right) \pm \sqrt {{{\left( 4 \right)}^2} - 4\left( 1 \right)\left( { - 3} \right)} }}{{2\left( 1 \right)}}\],
Now simplifying we get,
\[ \Rightarrow x = \dfrac{{ - 4 \pm \sqrt {16 - \left( { - 12} \right)} }}{2}\],
Now again simplifying we get,
\[ \Rightarrow x = \dfrac{{ - 4 \pm \sqrt {16 + 12} }}{2}\],
Further simplification we get,
\[ \Rightarrow x = \dfrac{{ - 4 \pm \sqrt {28} }}{2}\],
Now taking the square root we get,
\[ \Rightarrow x = \dfrac{{ - 4 \pm 2\sqrt 7 }}{2}\],
Now taking out the common terms we get,
\[ \Rightarrow x = - 2 \pm \sqrt 7 \],
So, the values of \[x\] will be \[ - 2 \pm \sqrt 7 \], and this is its simplest form of radical.

If we solve the given equation, i.e., \[{x^2} + 4x + 4 = 7\], then the values of $x$ are \[ - 2 + \sqrt 7 \] and \[ - 2 \pm \sqrt 7 \], and they are in their simplest form of radical.

Note:
Quadratic equation formula is a method of solving quadratic equations, but we should keep in mind that we can also solve the equation using completely the square, and we can cross check the values by using the above formula. Also we should always convert the coefficient of, to easily solve the equation by this method, and there are other methods to solve such kinds of solutions, another method used to solve the quadratic equation is by factoring method, in this method we should obtain the solution factorising quadratic equation terms. In these type of questions, we can solve by using quadratic formula i.e.,\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].