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Solve 3x² + 4x - 3 = 0 Using the Completing the Square Method
Mathematics
Grade 10 (Junior High School)
Question Content
Solve the quadratic equation 3x² + 4x - 3 = 0 by completing the square.
Correct Answer
x = -2/3 + √13/3 or x = -2/3 - √13/3
Detailed Solution Steps
1
Step 1: Isolate the constant term by adding 3 to both sides of the equation: 3x² + 4x = 3
2
Step 2: Divide every term by the coefficient of x² (which is 3) to make the leading coefficient 1: x² + (4/3)x = 1
3
Step 3: Complete the square on the left side. Take half of the coefficient of x, which is (4/3)÷2 = 2/3, square it to get (2/3)² = 4/9. Add this value to both sides of the equation: x² + (4/3)x + 4/9 = 1 + 4/9
4
Step 4: Simplify both sides. The left side becomes a perfect square trinomial, and the right side is a single fraction: (x + 2/3)² = 13/9
5
Step 5: Take the square root of both sides, remembering to include both the positive and negative square roots: x + 2/3 = ±√(13/9) = ±√13/3
6
Step 6: Solve for x by subtracting 2/3 from both sides: x = -2/3 ± √13/3
Knowledge Points Involved
1
Completing the Square for Quadratic Equations
A method to rewrite a quadratic expression ax²+bx+c into the form a(x-h)²+k, where (h,k) is the vertex of the parabola. For solving equations, it transforms the quadratic into a perfect square trinomial plus a constant, allowing solution via square roots. Used when factoring is not straightforward, and it is the basis for deriving the quadratic formula.
2
Properties of Square Roots
When taking the square root of both sides of an equation, both the positive and negative square roots must be considered because squaring either a positive or negative number results in a positive value. For any positive real number n, √n² = |n|, so if x² = n, then x = ±√n.
3
Operations with Fractions
Includes finding common denominators to add or subtract fractions, and simplifying complex fractions. Critical in completing the square when adjusting the constant term to balance the equation, as it requires combining fractional values accurately.
4
Quadratic Equation Standard Form
The standard form of a quadratic equation is ax²+bx+c=0, where a≠0. Before completing the square, the leading coefficient a must be 1, so all terms are divided by a if a≠1. This form is the starting point for all quadratic solution methods: factoring, completing the square, quadratic formula, and graphing.
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