1)Use completing the square method to solve the equation; 3x^2-7x+2=0
2)Solve the equation; 6a=5+4/a using completing the square method
3)A quadratic equation x^2+ax-b=0 has roots 1 and -5.Find the values of a and b
4)Find the quadratic function in x that cuts the x-axis at[-7/2,0] and [5/2,0].
5)Given the roots of a quadratic equation in x are 1.5+√3 and 1.5-√3, find the equation in its simplest form
6)The roots of an equation ax^2+bx+c=0 are 2.5+or-√2.Find the values of a,b and c
7)Given that 9x^2+36x-40+k is a perfect square,find the value of k
8)The equation of a curve is given as y+4=(x-1)^2.State the x and y intercepts of the curve
9)Find the coordinates of the points of intersection of the straight line y=x+5 and the hyperbola xy=-6
10)Solve the simultaneous equation; x^2+y^2=26
x-y=4
11)Solve for x in the equation 3x^2+18x+15=0 using completing the square methods
13)Determine the value of k if the expression 16x^2-8kx+25 is a perfect square
15)Find the points where the parabola y=x^2+4x+3 intersects with the straight line y=-x+9
18)Given that the discriminant of the equation 4x^2-10x+k=0 is 0,find k
1) To solve the equation 3x^2 - 7x + 2 = 0 using the completing the square method, we can follow these steps:
Start with the equation: 3x^2 - 7x + 2 = 0
Move the constant term (2) to the right side: 3x^2 - 7x = -2
Divide the entire equation by the coefficient of x^2 (3): x^2 - (7/3)x = -2/3
To complete the square, take half of the coefficient of x (-7/3), square it, and add it to both sides of the equation:
x^2 - (7/3)x + (-7/6)^2 = -2/3 + (-7/6)^2
Simplify:
x^2 - (7/3)x + 49/36 = -2/3 + 49/36
x^2 - (7/3)x + 49/36 = -24/36 + 49/36
x^2 - (7/3)x + 49/36 = 25/36
Factor the perfect square trinomial on the left side:
(x - 7/6)^2 = 25/36
Take the square root of both sides:
x - 7/6 = ± √(25/36)
Simplify the square root:
x - 7/6 = ± 5/6
Solve for x in both cases:
Case 1: x - 7/6 = 5/6
x = 7/6 + 5/6
x = 12/6
x = 2
Case 2: x - 7/6 = -5/6
x = 7/6 - 5/6
x = 2/6
x = 1/3
Therefore, the solutions to the equation 3x^2 - 7x + 2 = 0 are x = 2 and x = 1/3.
2) To solve the equation 6a = 5 + 4/a using the completing the square method, we can follow these steps:
Start with the equation: 6a = 5 + 4/a
Move all terms to one side to create a quadratic equation: 6a - 5 - 4/a = 0
To complete the square, multiply both sides of the equation by a:
6a^2 - 5a - 4 = 0
To complete the square, take half of the coefficient of a (-5/2), square it, and add it to both sides of the equation:
6a^2 - 5a + (-5/2)^2 - (-5/2)^2 - 4 = 0
Simplify:
6a^2 - 5a + 25/4 - 25/4 - 4 = 0
Combine like terms:
6a^2 - 5a + 9/4 = 0
Factor the perfect square trinomial on the left side:
(2a - 3/2)^2 = 0
Take the square root of both sides:
2a - 3/2 = ± √(0)
Simplify the square root:
2a - 3/2 = 0
Solve for a:
Case 1: 2a - 3/2 = 0
2a = 3/2
a = 3/4
Therefore, the solution to the equation 6a = 5 + 4/a is a = 3/4.
3) We are given that the quadratic equation x^2 + ax - b = 0 has roots 1 and -5.
The sum of the roots of a quadratic equation can be found using the formula -b/a, where a is the coefficient of x^2 and b is the coefficient of x. The product of the roots can be found using the formula c/a, where c is the constant term.
In this case, the sum of the roots is 1 + (-5) = -4, and the product of the roots is 1 * (-5) = -5.
Since the sum of the roots is -4, we can set the coefficient of x equal to -4. Therefore, a = -4.
Since the product of the roots is -5, we can set the constant term equal to -5. Therefore, b = -5.
Therefore, the values of a and b are a = -4 and b = -5.
4) To find the quadratic function in x that cuts the x-axis at [-7/2,0] and [5/2,0], we can use the fact that the x-intercepts occur when y = 0.
Given that the x-intercepts are -7/2 and 5/2, we can set up the equation:
(x + 7/2)(x - 5/2) = 0
Expand the equation:
x^2 + (7/2)x - (5/2)x - 35/4 = 0
Combine like terms:
x^2 + (2/2)x - 35/4 = 0
Simplify:
x^2 + (1/2)x - 35/4 = 0
Therefore, the quadratic function in x that cuts the x-axis at [-7/2,0] and [5/2,0] is f(x) = x^2 + (1/2)x - 35/4.
5) Given the roots 1.5 + √3 and 1.5 - √3, we can find the equation in its simplest form by using the fact that the sum and product of the roots of a quadratic equation are related to its coefficients.
The sum of the roots can be found by taking -b/a, and the product of the roots can be found by taking c/a.
In this case, the sum of the roots is (1.5 + √3) + (1.5 - √3) = 3.
The product of the roots is (1.5 + √3)(1.5 - √3) = 2.25 - 3 = -0.75.
Since the sum of the roots is 3, we can set the coefficient of x equal to -3. Therefore, a = -3.
Since the product of the roots is -0.75, we can set the constant term equal to -0.75. Therefore, b = -0.75.
Therefore, the equation in its simplest form is f(x) = -3x^2 - 0.75.
6) We are given that the roots of the equation ax^2 + bx + c = 0 are 2.5 + √2 and 2.5 - √2.
The sum of the roots can be found by taking -b/a, and the product of the roots can be found by taking c/a.
In this case, the sum of the roots is (2.5 + √2) + (2.5 - √2) = 5.
The product of the roots is (2.5 + √2)(2.5 - √2) = (2.5)^2 - (√2)^2 = 6.25 - 2 = 4.25.
Since the sum of the roots is 5, we can set the coefficient of x equal to -5. Therefore, b = -5.
Since the product of the roots is 4.25, we can set the constant term equal to 4.25. Therefore, c = 4.25.
Since we don't have the value of a, we can express it in terms of the known coefficients:
a = c/(b - 5)
Substituting the values of b and c:
a = 4.25/(-5 - 5) = 4.25/-10 = -0.425
Therefore, the values of a, b, and c are a = -0.425, b = -5, and c = 4.25.
7) We are given that 9x^2 + 36x - 40 + k is a perfect square.
To determine the value of k, we can use the fact that the square of a term can be expressed as the square of its binomial, such as (a + b)^2 = a^2 + 2ab + b^2.
Since the given expression is a perfect square, we can rewrite it as:
(3x + p)^2 = 9x^2 + 6px + p^2
Comparing the coefficients of the original expression and the rewritten expression, we have:
9x^2 + 36x - 40 + k = 9x^2 + 6px + p^2
From this, we can find that p = 6, as the coefficient of x in the original expression is 36, which equals 6px.
Substituting p = 6 back into the rewritten expression:
(3x + 6)^2 = 9x^2 + 36x + 36
Expanding the squared term:
9x^2 + 36x + 36 = 9x^2 + 36x - 40 + k
We can see that the x^2 and x terms cancel out:
36 = -40 + k
k = 36 + 40
k = 76
Therefore, the value of k is 76.
8) The equation of the curve, stated as y + 4 = (x - 1)^2, can also be written as y = (x - 1)^2 - 4.
To find the x-intercepts, we set y = 0 and solve for x:
0 = (x - 1)^2 - 4
Simplify the equation:
4 = (x - 1)^2
Take the square root of both sides:
±√4 = ±(x - 1)
Simplify:
±2 = ±(x - 1)
Solve for x in both cases:
Case 1: x - 1 = 2
x = 2 + 1
x = 3
Case 2: x - 1 = -2
x = -2 + 1
x = -1
Therefore, the x-intercepts are x = 3 and x = -1.
To find the y-intercept, we set x = 0 and solve for y:
y = (0 - 1)^2 - 4
Simplify:
y = 1 - 4
y = -3
Therefore, the y-intercept is y = -3.
Therefore, the x-intercept is (3, 0) and (-1, 0), and the y-intercept is (0, -3).
9) To find the coordinates of the points of intersection between the straight line y = x + 5 and the hyperbola xy = -6, we can substitute y in the hyperbola equation with x + 5.
xy = -6 can be rewritten as x(x + 5) = -6.
Expanding the equation:
x^2 + 5x = -6
Rearranging the equation and making it equal to zero:
x^2 + 5x + 6 = 0
Factor the quadratic equation:
(x + 3)(x + 2) = 0
Solve for x in both cases:
Case 1: x + 3 = 0
x = -3
Case 2: x + 2 = 0
x = -2
Substitute the values of x back into y = x + 5:
When x = -3:
y = -3 + 5
y = 2
When x = -2:
y = -2 + 5
y = 3
Therefore, the coordinates of the points of intersection are (-3, 2) and (-2, 3).
10) To solve the simultaneous equations x^2 + y^2 = 26 and x - y = 4, we can use the substitution method:
Rearrange the second equation to solve for x:
x = y + 4
Substitute this value of x into the first equation:
(y + 4)^2 + y^2 = 26
Expand and simplify the equation:
y^2 + 8y + 16 + y^2 = 26
2y^2 + 8y + 16 - 26 = 0
2y^2 + 8y - 10 = 0
y^2 + 4y - 5 = 0
Factor the quadratic equation:
(y + 5)(y - 1) = 0
Solve for y in both cases:
Case 1: y + 5 = 0
y = -5
Case 2: y - 1 = 0
y = 1
Substitute these values of y back into the second equation to solve for x:
When y = -5:
x - (-5) = 4
x + 5 = 4
x = -1
When y = 1:
x - 1 = 4
x = 5
Therefore, the solutions to the simultaneous equations are (-1, -5) and (5, 1).
11) To solve for x in the equation 3x^2 + 18x + 15 = 0 using the completing the square method, we can follow these steps:
Start with the equation: 3x^2 + 18x + 15 = 0
Divide the entire equation by the coefficient of x^2 (3): x^2 + 6x + 5 = 0
To complete the square, take half of the coefficient of x (6), square it, and add it to both sides of the equation:
x^2 + 6x + 9 = - 5 + 9
Simplify:
(x + 3)^2 = 4
Take the square root of both sides:
x + 3 = ±√(4)
Simplify the square root:
x + 3 = ±2
Solve for x in both cases:
Case 1: x + 3 = 2
x = 2 - 3
x = -1
Case 2: x + 3 = -2
x = -2 - 3
x = -5
Therefore, the solutions to the equation 3x^2 + 18x + 15 = 0 are x = -1 and x = -5.
13) To determine the value of k if the expression 16x^2 - 8kx + 25 is a perfect square, we can use the fact that the square of a term can be expressed as the square of its binomial, such as (a + b)^2 = a^2 + 2ab + b^2.
In this case, we need to rewrite the expression in the form of a perfect square trinomial.
16x^2 - 8kx + 25 = (4x - a)^2
Expanding the perfect square trinomial:
(4x - a)^2 = 16x^2 - 8ax + a^2
Comparing the coefficients of the two expressions, we have:
16x^2 - 8kx + 25 = 16x^2 - 8ax + a^2
From this, we can find that -k = -8a and 25 = a^2.
To find the value of k, we need a value for a. If we assume a = 5, then from -k = -8a, we can determine that k = 8 * 5 = 40.
Therefore, the value of k is 40.
15) To find the points where the parabola y = x^2 + 4x + 3 intersects with the straight line y = -x + 9, we can set the two equations equal to each other and solve for x:
x^2 + 4x + 3 = -x + 9
Rearranging the equation and making it equal to zero:
x^2 + 5x + 6 = 0
Factor the quadratic equation:
(x + 3)(x + 2) = 0
Solve for x in both cases:
Case 1: x + 3 = 0
x = -3
Case 2: x + 2 = 0
x = -2
Substitute the values of x back into the equation y = -x + 9 to find the corresponding y-values:
When x = -3:
y = -(-3) + 9
y = 3 + 9
y = 12
When x = -2:
y = -(-2) + 9
y = 2 + 9
y = 11
Therefore, the points of intersection are (-3, 12) and (-2, 11).
18) The discriminant of the equation 4x^2 - 10x + k = 0 is 0.
The discriminant can be calculated using the formula b^2 - 4ac.
In this case, a = 4, b = -10, and the