Find The Zeros Of $y = X^2 - 8x - 3$ By Completing The Square.A. $x = \pm 3$ B. $x = 4 \pm \sqrt{19}$ C. $x = 3 \pm \sqrt{7}$ D. $x = -4 \pm \sqrt{19}$

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Introduction

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Completing the square is a powerful technique used to solve quadratic equations of the form $y = ax^2 + bx + c$. This method involves manipulating the equation to express it in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this article, we will use completing the square to find the zeros of the quadratic equation $y = x^2 - 8x - 3$.

Step 1: Write the Equation in General Form

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The given equation is already in general form: $y = x^2 - 8x - 3$. The first step in completing the square is to ensure that the coefficient of $x^2$ is 1. In this case, the coefficient is already 1, so we can proceed to the next step.

Step 2: Move the Constant Term to the Right-Hand Side

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To complete the square, we need to move the constant term to the right-hand side of the equation. We do this by subtracting $-3$ from both sides of the equation:

$$y + 3 = x^2 - 8x$$

Step 3: Add and Subtract the Square of Half the Coefficient of $x$

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The coefficient of $x$ is $-8$. To complete the square, we need to add and subtract the square of half the coefficient of $x$. Half of $-8$ is $-4$, and the square of $-4$ is $16$. We add and subtract $16$ from both sides of the equation:

$$y + 3 + 16 = x^2 - 8x + 16$$

Step 4: Factor the Left-Hand Side as a Perfect Square

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The left-hand side of the equation is now a perfect square:

$$(y + 3 + 16) = (x - 4)^2$$

Step 5: Simplify the Right-Hand Side

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We can simplify the right-hand side of the equation by evaluating the expression $y + 3 + 16$:

$$(y + 19) = (x - 4)^2$$

Step 6: Take the Square Root of Both Sides

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To find the zeros of the equation, we need to take the square root of both sides of the equation. We do this by taking the square root of the left-hand side and the right-hand side:

$$\sqrt{(y + 19)} = \sqrt{(x - 4)^2}$$

Step 7: Simplify the Square Root

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We can simplify the square root by evaluating the expression $\sqrt{(x - 4)^2}$:

$$\sqrt{(y + 19)} = |x - 4|$$

Step 8: Solve for $x$

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To find the zeros of the equation, we need to solve for $x$. We do this by setting the expression $\sqrt{(y + 19)}$ equal to zero and solving for $x$:

$$\sqrt{(y + 19)} = 0$$

$$y + 19 = 0$$

$$y = -19$$

Substituting $y = -19$ into the original equation, we get:

$$-19 = x^2 - 8x - 3$$

$$x^2 - 8x - 22 = 0$$

Step 9: Solve the Quadratic Equation

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We can solve the quadratic equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -8$, and $c = -22$. Substituting these values into the quadratic formula, we get:

$$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-22)}}{2(1)}$$

$$x = \frac{8 \pm \sqrt{64 + 88}}{2}$$

$$x = \frac{8 \pm \sqrt{152}}{2}$$

$$x = \frac{8 \pm 4\sqrt{19}}{2}$$

$$x = 4 \pm 2\sqrt{19}$$

Conclusion

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In this article, we used completing the square to find the zeros of the quadratic equation $y = x^2 - 8x - 3$. We started by writing the equation in general form and then moved the constant term to the right-hand side. We then added and subtracted the square of half the coefficient of $x$ and factored the left-hand side as a perfect square. Finally, we took the square root of both sides and solved for $x$ using the quadratic formula. The zeros of the equation are $x = 4 \pm 2\sqrt{19}$.

Final Answer

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The final answer is: $\boxed{4 \pm 2\sqrt{19}}$

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Introduction

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Completing the square is a powerful technique used to solve quadratic equations of the form $y = ax^2 + bx + c$. In our previous article, we used completing the square to find the zeros of the quadratic equation $y = x^2 - 8x - 3$. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

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A: Completing the square is a technique used to solve quadratic equations by manipulating the equation to express it in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: When should I use completing the square?

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A: You should use completing the square when you are given a quadratic equation in general form and you want to find the zeros of the equation. Completing the square is particularly useful when the equation is not easily factorable.

Q: How do I know if an equation can be solved by completing the square?

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A: An equation can be solved by completing the square if it is in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. If the equation is not in this form, you may need to use a different technique to solve it.

Q: What are the steps involved in completing the square?

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A: The steps involved in completing the square are:

1. Write the equation in general form.
2. Move the constant term to the right-hand side.
3. Add and subtract the square of half the coefficient of $x$.
4. Factor the left-hand side as a perfect square.
5. Take the square root of both sides.
6. Solve for $x$.

Q: What is the vertex of a parabola?

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A: The vertex of a parabola is the point $(h, k)$ where the parabola changes direction. The vertex can be found by completing the square and expressing the equation in the form $y = a(x - h)^2 + k$.

Q: How do I find the zeros of a quadratic equation using completing the square?

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A: To find the zeros of a quadratic equation using completing the square, you need to follow the steps outlined above. Once you have expressed the equation in the form $y = a(x - h)^2 + k$, you can take the square root of both sides and solve for $x$.

Q: What are some common mistakes to avoid when completing the square?

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A: Some common mistakes to avoid when completing the square include:

• Not moving the constant term to the right-hand side.
• Not adding and subtracting the square of half the coefficient of $x$.
• Not factoring the left-hand side as a perfect square.
• Not taking the square root of both sides.
• Not solving for $x$ correctly.

Q: Can completing the square be used to solve all types of quadratic equations?

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A: No, completing the square can only be used to solve quadratic equations that are in the form $y = ax^2 + bx + c$. If the equation is not in this form, you may need to use a different technique to solve it.

Q: Is completing the square a difficult technique to learn?

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A: Completing the square is a relatively simple technique to learn, but it does require some practice to become proficient. With patience and persistence, you can master the technique and use it to solve a wide range of quadratic equations.

Conclusion

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In this article, we have answered some frequently asked questions about completing the square. We have discussed the steps involved in completing the square, the vertex of a parabola, and some common mistakes to avoid. We have also provided some tips and advice for learning and mastering the technique. With this knowledge, you should be able to complete the square with confidence and solve a wide range of quadratic equations.

Final Answer

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The final answer is: Completing the square is a powerful technique used to solve quadratic equations by manipulating the equation to express it in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.