Question

Solve ${{x}^{2}}-3x+2=0$?

Answer 1

Hint:
During this problem we want to solve the given quadratic equation i.e., we would like to calculate the values of $x$ where the given equation is satisfied. We have several methods to solve a quadratic equation. But in for this problem, we are going to use the quadratic formula which is given by $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.To use the above formula we want to possess the quadratic equation in standard form which is $a{{x}^{2}}+bx+c=0$, so we will simplify the given equation and convert it into the standard form. After that we will compare the given equation with the standard quadratic equation $a{{x}^{2}}+bx+c=0$ and write the values of $a$, $b$, $c$. Now we will substitute those values in the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ and simplify the obtained equation to get the required result.

Formula Used:
The roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ is given by $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.

Complete step by step solution:
Given equation ${{x}^{2}}-3x+2=0$.
Comparing the above quadratic equation with standard quadratic equation $a{{x}^{2}}+bx+c=0$, then we will get the values of $a$, $b$, $c$ as
$a=1$, $b=-3$, $c=2$.
To find the roots of the quadratic equation we have the formula
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substituting the values of $a$, $b$, $c$ in the above equation, then we will get
$\Rightarrow x=\dfrac{-\left( -3 \right)\pm \sqrt{{{\left( -3 \right)}^{2}}-4\left( 1 \right)\left( 2 \right)}}{2\left( 1 \right)}$
We know that when we multiplied a negative sign with the negative sign, then we will get a positive sign. Applying the above rule and simplifying the above equation, then we will get
$
   \Rightarrow x=\dfrac{3\pm \sqrt{9-8}}{2} \\
  \Rightarrow x=\dfrac{3\pm \sqrt{1}}{2} \\
$
In the above equation we have the value $\sqrt{1}$. We know that $\sqrt{1}=1$. Substituting this value in the above equation, then we will get
$
   x=\dfrac{3\pm 1}{2} \\
  \Rightarrow x=\dfrac{3+1}{2}\text{ or }\dfrac{3-1}{2} \\
  \Rightarrow x=\dfrac{4}{2}\text{ or }\dfrac{2}{2} \\
  \Rightarrow x=2\text{ or 1} \\
$

Hence the roots of the given quadratic equation ${{x}^{2}}-3x+2=0$ are $1,2$.

Note:
We can also plot the graph of the given equation to calculate the roots. When we plot the given equation in graph paper, then we will get

From the above graph also, we can write that the roots of the given equation are $1,2$.