SAT Math: Most common Q&A's

Find the slope of the line passing through the points (1, 2) and (3, 6).

Solution: The slope of a line is the change in y over the change in x, which is calculated by dividing the difference in y-coordinates by the difference in x-coordinates. The slope of the line passing through (1, 2) and (3, 6) is (6 - 2) / (3 - 1) = 4. The answer is 4.

1. Solve for x in the equation x^2 - 4x + 4 = 0.

Solution: To solve for x in this quadratic equation, we can use the quadratic formula, which states that x = (-b ± √(b^2 - 4ac)) / 2a. Plugging in the values from the equation, we get x = (4 ± √(4^2 - 4 * 1 * 4)) / 2 * 1 = (4 ± √(16 - 16)) / 2 = (4 ± 0) / 2 = 2. The answer is 2.

1. Find the equation of the line passing through the point (3, 5) and perpendicular to the line y = 2x + 1.

Solution: To find the equation of a line that passes through a point and is perpendicular to another line, we can use the slope-point form of a line, which states that the equation of a line with slope m passing through the point (x1, y1) is y - y1 = m(x - x1). To find the slope of the perpendicular line, we need to find the negative reciprocal of the slope of the given line, which is 2. The negative reciprocal of 2 is -0.5. The equation of the line passing through (3, 5) with a slope of -0.5 is y - 5 = -0.5(x - 3). The answer is y - 5 = -0.5x + 4.5.

1. Simplify the expression √(36) + √(49) - √(64).

Solution: To simplify this expression, we need to find the exact value of each square root. √(36) = 6, √(49) = 7, and √(64) = 8. Plugging in these values, we get 6 + 7 - 8 = 5. The answer is 5.

1. Solve for x in the equation log4 (x^2) = 2.

Solution: To solve for x in this logarithmic equation, we need to use the change of base formula, which states that logb a = logc a / logc b. In this case, log4 (x^2) = 2, so we have x^2 = 4^2 = 16. To find the value of x, we can take the square root of 16 to get x = ± 4. The answer is ± 4.

1. Simplify the expression (3x^2 - 4x + 1) / (2x^2 - 8x + 4).

Solution: To simplify this expression, we need to perform polynomial division. The answer is 3/2 x - 1 + 1 / 2x - 2.

1. Find the equation of the circle with center (3, 4) and radius 2.

Solution: The equation of a circle with center (h, k) and radius r is (x - h)

1. What is the value of x in the equation 3x + 5 = 16?

Solution: Subtract 5 from both sides to isolate 3x, 3x = 11. Then divide both sides by 3 to solve for x, x = 11/3 or approximately 3.67.

1. Simplify the expression √144.

Solution: √144 can be written as √(12^2), and since 12^2 = 144, √144 = 12.

1. Solve for x in the equation x^2 + 4x - 5 = 0.

Solution: Factoring the equation, (x - 1)(x + 5) = 0. So either x - 1 = 0 or x + 5 = 0, which means x = 1 or x = -5.

1. If a circle has a diameter of 12, what is its circumference?

Solution: The circumference of a circle can be found by multiplying its diameter by π. So, the circumference of a circle with a diameter of 12 is 12 * π, which is approximately 37.7.

1. Simplify the expression (2x + 3)^2.

Solution: Using the exponent rule, (2x + 3)^2 = 4x^2 + 12x + 9.

1. What is the equation of the line that passes through the points (3, 4) and (7, 10)

Solution: To find the equation of the line, we need to find the slope and the y-intercept. Using the slope formula, the slope m is (10-4)/(7-3) = 3. Then using the point-slope form of a line, the equation is y - 4 = 3(x - 3), or y = 3x + 6.

1. What is the equation of the circle with center (4, 5) and a radius of 3?

Solution: The equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. So, the equation of the circle with center (4, 5) and radius 3 is (x - 4)^2 + (y - 5)^2 = 9.

1. What is the length of the line segment connecting the points (-2, 3) and (4, 7)?

Solution: Using the distance formula, the length of the line segment is √((4 - (-2))^2 + (7 - 3)^2) = √(6^2 + 4^2) = √(36 + 16) = √52.