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Solve $x^2 + 4x + 3 = 0$ by Completing the Square Step-by-Step
Mathematics
Grade 9 (Junior High School)
Question Content
Solve the equation $x^2 + 4x + 3 = 0$ by completing the square. Drag numbers to the lines to complete the solution to the equation. Step 1: $x^2 + 4x + 3 = 0$ Step 2: $(x^2 + 4x + \\underline{\\quad}) + \\underline{\\quad} + 3 = 0$ Step 3: $(x + \\underline{\\quad})^2 = \\underline{\\quad}$ Step 4: $x + \\underline{\\quad} = \\pm \\underline{\\quad}$ Solution: $x = \\underline{\\quad}$ and $x = \\underline{\\quad}$ (Available numbers: -4, -3, -2, -1, 0, 1, 2, 3, 4)
Correct Answer
Step 2 blanks: 4, -4; Step 3 blanks: 2, 1; Step 4 blanks: 2, 1; Solution blanks: -1, -3
Detailed Solution Steps
1
Step 1: Start with the original quadratic equation: $x^2 + 4x + 3 = 0$
2
Step 2: To complete the square for $x^2 + 4x$, take half of the coefficient of $x$ (which is $4$), so $\\frac{4}{2}=2$, then square it: $2^2=4$. Add and subtract this value to keep the equation balanced: $(x^2 + 4x + 4) + (-4) + 3 = 0$
3
Step 3: Rewrite the perfect square trinomial as a squared binomial, then simplify the constant terms: $(x + 2)^2 - 1 = 0$, which rearranges to $(x + 2)^2 = 1$
4
Step 4: Take the square root of both sides of the equation, remembering to include both positive and negative roots: $x + 2 = \\pm 1$
5
Step 5: Solve for $x$ in two cases: Case 1: $x + 2 = 1$, so $x = 1 - 2 = -1$; Case 2: $x + 2 = -1$, so $x = -1 - 2 = -3$
Knowledge Points Involved
1
Completing the Square for Quadratic Expressions
For a quadratic expression of the form $x^2 + bx$, completing the square means adding $\\left(\\frac{b}{2}\\right)^2$ to create a perfect square trinomial $(x + \\frac{b}{2})^2$. This method is used to solve quadratic equations, convert quadratic functions to vertex form, and derive the quadratic formula. When using it to solve equations, you must subtract the same value you add to keep the equation equivalent.
2
Perfect Square Trinomials
A perfect square trinomial is a trinomial that can be written as the square of a binomial. The form is $x^2 + 2ax + a^2 = (x + a)^2$ or $x^2 - 2ax + a^2 = (x - a)^2$. In this problem, $x^2 + 4x + 4$ is a perfect square trinomial equal to $(x + 2)^2$.
3
Square Root Property of Equations
The square root property states that if $u^2 = k$ where $k \\geq 0$, then $u = \\sqrt{k}$ or $u = -\\sqrt{k}$, which is written as $u = \\pm \\sqrt{k}$. This is used to isolate the variable after creating a perfect square in the completing the square method.
4
Solving Quadratic Equations
Quadratic equations are equations of the form $ax^2 + bx + c = 0$ where $a \\neq 0$. Completing the square is one of three common methods to solve them, along with factoring and using the quadratic formula. It works for all quadratic equations, even those that cannot be factored over integers.
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