How do you complete the square for the equation $$x^2 - 8x - 3 = 0$$ ?

1. Move the constant to the other side: $$x^2 - 8x = 3$$ . 2. Take half of the coefficient of $$x$$ , square it: $$(-4)^2 = 16$$ . 3. Add 16 to both sides: $$x^2 - 8x + 16 = 19$$ . 4. Factor: $$(x - 4)^2 = 19$$ . 5. Solve: $$x = 4 \\pm \sqrt{19}$$ .

What is the solution to the equation $$n^2 - 18n - 40 = 0$$ using completing the square?

1. Move the constant: $$n^2 - 18n = 40$$ . 2. Half of -18 is -9, square it: $$(-9)^2 = 81$$ . 3. Add 81: $$n^2 - 18n + 81 = 121$$ . 4. Factor: $$(n - 9)^2 = 121$$ . 5. Solve: $$n = 9 \\pm 11$$ , giving solutions -2 and 20.

How do you find the value of $$c$$ that makes $$x^2 + 5x + c$$ a perfect square?

To make it a perfect square, take half of 5, square it: $$(2.5)^2 = 6.25$$ . Thus, $$c = 6.25$$ .

What is the general form of a quadratic equation?

The general form of a quadratic equation is $$ax^2 + bx + c = 0$$ , where $$a$$ , $$b$$ , and $$c$$ are constants.

What is the significance of the discriminant in a quadratic equation?

The discriminant, given by $$b^2 - 4ac$$ , determines the nature of the roots: if positive, there are two real roots; if zero, one real root; if negative, no real roots.

How can you check if a trinomial is a perfect square?

You can check if a trinomial is a perfect square by seeing if it can be factored into the form $$(a + b)^2$$ or $$(a - b)^2$$ .

What is the vertex form of a quadratic equation?

The vertex form of a quadratic equation is $$y = a(x - h)^2 + k$$ , where (h, k) is the vertex of the parabola.

What is the relationship between the coefficients and the roots of a quadratic equation?

The sum of the roots is equal to $$-b/a$$ and the product of the roots is equal to $$c/a$$ .

What is the purpose of completing the square in solving quadratic equations?

Completing the square allows us to rewrite the quadratic equation in a form that makes it easier to solve for the variable.

What is the formula for finding the roots of a quadratic equation using the quadratic formula?

The roots can be found using the formula $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$ .

How do you derive the quadratic formula from completing the square?

Start with the general form $$ax^2 + bx + c = 0$$ , divide by $$a$$ , move $$c/a$$ to the other side, complete the square, and solve for $$x$$ .

Completing the Square Flashcards

Student preview

15 questions

Show all answers

1.

FLASHCARD QUESTION

Front

What is the process of completing the square?

Back

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form of a perfect square trinomial, allowing for easier solution of the variable.

2.

FLASHCARD QUESTION

Front

What is a perfect square trinomial?

Back

3.

FLASHCARD QUESTION

Front

Back

4.

FLASHCARD QUESTION

Front

Back

5.

FLASHCARD QUESTION

Front

Back

6.

FLASHCARD QUESTION

Front

What is the first step in completing the square for any quadratic equation?

Back

The first step is to move the constant term to the other side of the equation.

7.

FLASHCARD QUESTION

Front

What does it mean to write solutions from least to greatest?

Back

It means to arrange the solutions in ascending order, starting with the smallest value.

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