Solve the quadratic equation: x3 - 4x + 3 = 0. Find its roots and vertex, and visualize the parabola curve on coordinate axes.
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Step-by-Step Solution
Step 1: Analyze the Equation
The given equation is x3−4x+3=0. This is a cubic equation, not a quadratic equation. A quadratic equation is of the form ax2+bx+c=0.
Step 2: Identify the Problem
The problem asks to solve a 'quadratic equation' and find its 'vertex' and visualize a 'parabola curve'. However, the provided equation x3−4x+3=0 is a cubic equation. Therefore, some parts of the request are not applicable to the given equation.
Step 3: Address Misidentification (if possible)
Since the equation is cubic, it does not represent a parabola, and it does not have a single vertex like a parabola. We can only find its roots. If there was a typo and a quadratic equation was intended, such as x2−4x+3=0, we would proceed differently. Given the cubic form, we will just find its roots.
Step 4: Find Roots by Factoring (Trial and Error)
Let P(x)=x3−4x+3. We look for integer roots using the Rational Root Theorem. Possible rational roots are factors of the constant term (3), which are ±1,±3.
Step 5: Test x=1
Substituting x=1 into the polynomial P(x), we get P(1)=1−4+3=0. Since P(1)=0, x=1 is a root of the equation, which means (x−1) is a factor of P(x).
Step 6: Perform Polynomial Division
Now we divide P(x) by (x−1) to find the remaining quadratic factor. Using polynomial long division or synthetic division, we find that (x3−4x+3)=(x−1)(x2+x−3).
Step 7: Find Roots of the Quadratic Factor
We need to find the roots of the quadratic equation x2+x−3=0. We will use the quadratic formula, where a=1, b=1, and c=−3.
Step 8: Apply Quadratic Formula
Substituting the values of a, b, and c into the quadratic formula, we get the remaining two roots for x. These roots are 2−1+13 and 2−1−13.
Step 9: List All Roots
The cubic equation x3−4x+3=0 has three roots: one integer root and two irrational roots obtained from the quadratic factor.
Step 10: Visualize the Cubic Function
Since the given equation is cubic, not quadratic, we will visualize the cubic function y=x3−4x+3 on the coordinate axes. This will show its general shape and where it crosses the x -axis (the roots).