Can anybody tell me step wise step how to solve this quadratic equation?
x^3-3x^2+4=0
Please tell me step wise step as i haven't solved such type of equations before.
Thank You
Even you subject is "Quadratic Equations", the problem you submitted is a cubic equation, and there appears to be no typo.
We will solve the cubic as it is posted.
x^3-3x^2+4=0
We note that the coefficients add up to zero if we reverse the sign of x^3, i.e. (-1)-3+4=0
This indicates that x=-1 is a zero.
Use long polynomial division to reduce the equation to:
(x^3-3x^2+4)/(x+1)=x²-4x+4
which is a quadratic equation.
We can solve by factorization:
x²-4x+4 = 0
(x-2)²=0
Therefore x=-1 or x=2(multiplicity=2), i.e. x=-1, x=2 or x=2.
any other easy method
I do not see an easier method except factoring trial and error. For this particular problem, you will probably get the answers by trial factoring quite easily.
Remember that this is a cubic equation, which actually has a solution formula more complicated than the above.
There is also the Newton-Ralphson approximation method by iterations.
The above methods (factoring, long division, quadratic equation) do not introduce any algebraic operation that is beyond secondary school level.
N+ ( n+2)=16
Sure! I'd be happy to guide you through the process of solving the quadratic equation x^3 - 3x^2 + 4 = 0 step-by-step.
Step 1: Identify the degree of the equation
In this case, the highest power of the variable (x) is 3, so this is a cubic equation, not a quadratic equation. To solve it, we need to look for different methods specific to cubic equations.
Step 2: Check for obvious roots
Start by checking if there are any obvious roots. In this case, try substituting small integer values (such as -2, -1, 0, 1, 2) into the equation and see if any of them make the equation true. If you find any such root, it will simplify the equation and make it easier to solve. However, in this example, there are no obvious roots.
Step 3: Apply Rational Root Theorem (optional)
If you still haven't found an obvious root, you can apply the Rational Root Theorem as an additional step. This theorem helps to identify potential rational roots (fractions) of the equation. Since the coefficient of the highest power term is 1 (x^3), any rational root (p/q) of the equation must satisfy the condition: p is a factor of 4 (the constant term) and q is a factor of 1 (the coefficient of x^3).
Step 4: Synthetic Division or Long Division (optional)
If you find a rational root from Step 3, you can use synthetic division or long division to divide the given equation by (x - r), where r is the rational root. By doing this, you will obtain a reduced polynomial equation of lower degree. If you don't find any rational roots, skip this step.
Step 5: Factor or Solve the Reduced Equation
After applying synthetic division or long division in Step 4, you will obtain a reduced equation with a lower degree. If it's still a quadratic equation, you can proceed to factor it using methods like factoring by grouping, completing the square, or applying the quadratic formula. If it's a linear equation (degree 1), solve it for the remaining root(s).
Step 6: Check for Complex Roots (optional)
Sometimes, cubic equations have complex roots involving imaginary numbers. If you haven't found all the roots in Step 5 and the degree of the reduced equation is greater than 2, you might need to apply methods like the Quadratic Formula for solving equations with complex roots.
Unfortunately, since the given equation is cubic, it is beyond the scope of a quadratic equation, and the general methods of solving quadratics won't be applicable. However, these steps will guide you through the general process of solving a quadratic equation.