Question
Question: How do you solve \(2{x^2} - x + 8 = 0\) ?...
How do you solve 2x2−x+8=0 ?
Solution
First compare the given quadratic equation to standard quadratic equation and find the value of numbers a, b and c in given equation. Then, substitute the values of a, b and c in the formula of discriminant and find the discriminant of the given equation. Finally, put the values of a, b and D in the roots of the quadratic equation formula and get the desired result.
Formula used:
The quantity D=b2−4ac is known as the discriminant of the equation ax2+bx+c=0 and its roots are given by
x=2a−b±D or x=2a−b±b2−4ac
Complete step by step answer:
We know that an equation of the form ax2+bx+c=0, a,b,c,x∈R, is called a Real Quadratic Equation.
The numbers a, b and c are called the coefficients of the equation.
The quantity D=b2−4ac is known as the discriminant of the equation ax2+bx+c=0 and its roots are given by
x=2a−b±D or x=2a−b±b2−4ac
So, first compare 2x2−x+8=0 quadratic equation to standard quadratic equation and find the value of numbers a, b and c.
Comparing 2x2−x+8=0 with ax2+bx+c=0, we get
a=2, b=−1 and c=8
Now, substitute the values of a, b and c in D=b2−4ac and find the discriminant of the given equation.
D=(−1)2−4(2)(8)
After simplifying the result, we get
⇒D=1−64
⇒D=−63
Which means the given equation has no real roots.
Now putting the values of a, b and D in x=2a−b±D, we get
x=2×2−(−1)±63i
It can be written as
⇒x=41±63i
⇒x=41+463i and x=41−463i
So, x=41+463i and x=41−463i are roots/solutions of equation 2x2−x+8=0.
Therefore, the solutions to the quadratic equation 2x2−x+8=0 are x=41+463i and x=41−463i..
Note: We can check whether x=41+463i and x=41−463i are roots/solutions of equation 2x2−x+8=0 by putting the value of x in given equation.
Putting x=41+463i in LHS of equation 2x2−x+8=0.
LHS=2(41+463i)2−(41+463i)+8
On simplification, we get
⇒LHS=2(161−1663+863i)−41−463i+8
⇒LHS=−431+463i+431−463i
⇒LHS=0
∴LHS=RHS
Thus, x=41+463i is a solution of equation 2x2−x+8=0.
Putting x=41−463i in LHS of equation 2x2−x+8=0.
LHS=2(41−463i)2−(41−463i)+8
On simplification, we get
⇒LHS=2(161−1663−863i)−41+463i+8
⇒LHS=−431−463i+431+463i
⇒LHS=0
∴LHS=RHS
Thus, x=41−463i is a solution of equation 2x2−x+8=0.
Final solution: Therefore, the solutions to the quadratic equation 2x2−x+8=0 are x=41+463i and x=41−463i.