
How do you solve \[{{x}^{2}}-6x-9=0\] by completing the square?
Answer
443.1k+ views
Hint: Assume the expression \[{{x}^{2}}-6x-9=y\] and compare it with the general form given as: - \[y=a{{x}^{2}}+bx+c\]. Find the respective values of a, b and c. Now, find the discriminant of the given quadratic equation by using the formula: - \[D={{b}^{2}}-4ac\], where D = discriminant. Now, write the expression as: - \[y=a\left[ {{\left( x+\dfrac{b}{2a} \right)}^{2}}-\dfrac{D}{4{{a}^{2}}} \right]\] and substitute it equal to 0 to find the two values of x.
Complete step by step solution:
Here, we have been provided with the quadratic equation: - \[{{x}^{2}}-6x-9=0\] and we are asked to solve it. That means we have to find the values of x. We have been asked to use completing the square method.
Now, we know that any quadratic equation of the form \[y=a{{x}^{2}}+bx+c\] can be simplified as \[y=a\left[ {{\left( x+\dfrac{b}{2a} \right)}^{2}}-\dfrac{D}{4{{a}^{2}}} \right]\], using completing the square method. Here, ‘D’ denotes the discriminant. So, on assuming \[{{x}^{2}}-6x-9=y\] and comparing it with the general quadratic equation, we get,
\[\Rightarrow \] a = 1, b = -6, c = -9
Applying the formula for discriminant of a quadratic equation given as, \[D={{b}^{2}}-4ac\], we get,
\[\begin{align}
& \Rightarrow D={{\left( -6 \right)}^{2}}-4\left( 1 \right)\left( -9 \right) \\
& \Rightarrow D=36+36 \\
& \Rightarrow D=72 \\
\end{align}\]
Therefore, substituting the values in the simplified form of y, we get,
\[\begin{align}
& \Rightarrow y=1\left[ {{\left( x+\left( \dfrac{-6}{2\times 1} \right) \right)}^{2}}-\dfrac{72}{4\times {{\left( 1 \right)}^{2}}} \right] \\
& \Rightarrow y=\left[ {{\left( x-3 \right)}^{2}}-18 \right] \\
\end{align}\]
Substituting y = 0, we get,
\[\Rightarrow \left[ {{\left( x-3 \right)}^{2}}-18 \right]=0\]
\[\begin{align}
& \Rightarrow {{\left( x-3 \right)}^{2}}-18=0 \\
& \Rightarrow {{\left( x-3 \right)}^{2}}=18 \\
\end{align}\]
Taking square root both the sides, we get,
\[\begin{align}
& \Rightarrow \left( x-3 \right)=\pm \sqrt{18} \\
& \Rightarrow x-3=\pm 3\sqrt{2} \\
& \Rightarrow x=3\pm 3\sqrt{2} \\
\end{align}\]
Considering the two signs one – by – one, we get,
\[\Rightarrow x=\left( 3+3\sqrt{2} \right)\] or \[x=\left( 3-3\sqrt{2} \right)\]
\[\Rightarrow x=3\left( 1+\sqrt{2} \right)\] or \[x=3\left( 1-\sqrt{2} \right)\]
Hence, the solutions of the given quadratic equation are: \[x=3\left( 1+\sqrt{2} \right)\] and \[x=3\left( 1-\sqrt{2} \right)\].
Note:
One may also use the quadratic formula to solve the question and check if we are getting the same values of x. Note that the general expression of this completing the square method is also known as the vertex form. Generally, this form is used in coordinate geometry of parabola to find the vertex of the parabola which is given as \[\left( \dfrac{-b}{2a},\dfrac{-D}{4a} \right)\]. Note that the quadratic formula: - \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] is derived from completing the square method. Here it would be difficult to use the middle term split method because it may be difficult to think of the factors like: \[\left[ x-\left( 3+3\sqrt{2} \right) \right]\] and \[\left[ x-\left( 3-3\sqrt{2} \right) \right]\].
Complete step by step solution:
Here, we have been provided with the quadratic equation: - \[{{x}^{2}}-6x-9=0\] and we are asked to solve it. That means we have to find the values of x. We have been asked to use completing the square method.
Now, we know that any quadratic equation of the form \[y=a{{x}^{2}}+bx+c\] can be simplified as \[y=a\left[ {{\left( x+\dfrac{b}{2a} \right)}^{2}}-\dfrac{D}{4{{a}^{2}}} \right]\], using completing the square method. Here, ‘D’ denotes the discriminant. So, on assuming \[{{x}^{2}}-6x-9=y\] and comparing it with the general quadratic equation, we get,
\[\Rightarrow \] a = 1, b = -6, c = -9
Applying the formula for discriminant of a quadratic equation given as, \[D={{b}^{2}}-4ac\], we get,
\[\begin{align}
& \Rightarrow D={{\left( -6 \right)}^{2}}-4\left( 1 \right)\left( -9 \right) \\
& \Rightarrow D=36+36 \\
& \Rightarrow D=72 \\
\end{align}\]
Therefore, substituting the values in the simplified form of y, we get,
\[\begin{align}
& \Rightarrow y=1\left[ {{\left( x+\left( \dfrac{-6}{2\times 1} \right) \right)}^{2}}-\dfrac{72}{4\times {{\left( 1 \right)}^{2}}} \right] \\
& \Rightarrow y=\left[ {{\left( x-3 \right)}^{2}}-18 \right] \\
\end{align}\]
Substituting y = 0, we get,
\[\Rightarrow \left[ {{\left( x-3 \right)}^{2}}-18 \right]=0\]
\[\begin{align}
& \Rightarrow {{\left( x-3 \right)}^{2}}-18=0 \\
& \Rightarrow {{\left( x-3 \right)}^{2}}=18 \\
\end{align}\]
Taking square root both the sides, we get,
\[\begin{align}
& \Rightarrow \left( x-3 \right)=\pm \sqrt{18} \\
& \Rightarrow x-3=\pm 3\sqrt{2} \\
& \Rightarrow x=3\pm 3\sqrt{2} \\
\end{align}\]
Considering the two signs one – by – one, we get,
\[\Rightarrow x=\left( 3+3\sqrt{2} \right)\] or \[x=\left( 3-3\sqrt{2} \right)\]
\[\Rightarrow x=3\left( 1+\sqrt{2} \right)\] or \[x=3\left( 1-\sqrt{2} \right)\]
Hence, the solutions of the given quadratic equation are: \[x=3\left( 1+\sqrt{2} \right)\] and \[x=3\left( 1-\sqrt{2} \right)\].
Note:
One may also use the quadratic formula to solve the question and check if we are getting the same values of x. Note that the general expression of this completing the square method is also known as the vertex form. Generally, this form is used in coordinate geometry of parabola to find the vertex of the parabola which is given as \[\left( \dfrac{-b}{2a},\dfrac{-D}{4a} \right)\]. Note that the quadratic formula: - \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] is derived from completing the square method. Here it would be difficult to use the middle term split method because it may be difficult to think of the factors like: \[\left[ x-\left( 3+3\sqrt{2} \right) \right]\] and \[\left[ x-\left( 3-3\sqrt{2} \right) \right]\].
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Explain the Treaty of Vienna of 1815 class 10 social science CBSE

Write an application to the principal requesting five class 10 english CBSE
