Find the roots of f(x) = 4x^2 + 2x - 7 using completing the square.
Start by setting the equation equal to zero. 4x^2 + 2x - 7 = 0 Then, divide everything by a (the number in front of x^2) x^2 + (1/2)x - (7/2) = 0 Then move c (the last number) to the other side. x^2 + (1/2)x = (7/2) Now take b (the number in front of the x), divide it by 2, square it, and add that number to both sides. (1/2)/2 = (1/2) x (2/1) = (2/2) = 1 1^2 = 1 x^2 + (1/2)x + 1 = (7/2) + 1 Now factor the left side. The trick is that we're always going to use b/2 so the left side will simplify to (x+(b/2))^2 At the same time, simplify the right side by first finding a common denominator and then adding fractions as usual. (x+1)^2 = (7/2) + (2/2) (x+1)^2 = (9/2) Take the square root of both sides. x+1 = (+/-) sqrt (9/2) Get x by itself by moving the number to the right. x = -1 (+/-) sqrt (9/2) Notice that 9 is a perfect square so we can simplify the top of the sqrt to 3. x = -1 (+/-) 3/(sqrt 2) You can never have a square root in the denominator so we must rationalize by multiplying the numerator and the denominator by sqrt 2. x = -1 (+/-) ((3 sqrt 2)/2)
Say we have a square inscribed in a circle. What is the arc that two legs of the square will make with the circle?
Since the square is inscribed in the circle, we know that each vertex of the square touches the circle. We also know that each angle in a square measures 90 degrees by definition of a square. Because the vertex of the square is touching the circle, this forms an inscribed angle. We know that the arc across from an inscribed angle is the same measure as the angle itself. Thus, because the angle measures 90 degrees, so does the arc.
If f(x) = x^2 + 2x + 1, find the roots of f(x) algebraically.
Roots are where the graph f(x) crosses the x-axis. However, since this problem specifies "algebraically," we cannot use a graph to solve this problem. We will start by setting f(x) = 0. Then, we will factor using trinomial factoring (what multiplies to the last number but adds to the middle number). We will then set each factor equal to zero and solve. x^2 + 2x + 1 = 0 (x+1)(x+1) = 0 This is because 1 times 1 is 1 (the last number) and 1+1 is 2 (the middle number) x+1 = 0 x+1 = 0 x = -1 x = -1 So here, we only have one root at x = -1.