Question
What are the solutions of the quadratic equation 
?
- x = -1 and x = 3
- x = 1 and x = -3
- x = 1 and x = 3
Hint:
Hint:-
The word "quadratic" is originated from the word "quad" and its meaning is "square". It means the quadratic equation has a variable raised to 2 as the greatest power term. The standard form of a quadratic equation is given by the equation ax2 + bx + c = 0, where a ≠ 0. We know that any value(s) of x that satisfies the equation is known as a solution (or) root of the equation and the process of finding the values of x which satisfies the equation ax2 + bx + c = 0 is known as solving quadratic equations.
The correct answer is: x = -1 and x = 3
- The given equation is 4x2-8x-12=0.
- We can see 4 is the like factor of all terms in the equation . So we can write the equation as
4 (x2- 2x - 3 )= 0
Factoring x2 - 2x - 3
The first term is, x2 its coefficient is 1 .
The middle term is, -2x its coefficient is -2 .
The last term i.e. constant is -3
Step-1 : Multiply the coefficient of the first term by the constant 1 × (-3) = -3
Step-2 : Find two factors of -3 whose sum equals the coefficient of the middle term, which is -2 .
-3 + 1 = -2
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -3 and 1
x2 - 3x + 1x - 3
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-3)
Add up the last 2 terms, pulling out common factors :
1 • (x-3)
Step-5 : Add up the four terms of step 4 :
(x+1) • (x-3)
Which is the desired factorization
- So, the equation becomes
4 (x + 1) (x - 3) = 0
- A product of several terms equals zero. When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately . In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solve : 4 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solve : x+1 = 0
Subtract 1 from both sides of the equation :
x = -1
Solve : x - 3 = 0
Add 3 to both sides of the equation :
x = 3
- Therefore, we get the roots of the equation as x = -1 and x = 3.
Note:-There are different ways of solving quadratic equations.
- Solving quadratic equations by factoring
- Solving quadratic equations by completing the square
- Solving quadratic equations by graphing
- Solving quadratic equations by quadratic formula
But most popular method is solving quadratic equations by factoring.
Related Questions to study
In the xy-plane, the point (2, 5) lies on the graph of the function f. If f(x)= k - x2 , where k is a constant, what is the value of k ?
In the xy-plane, the point (2, 5) lies on the graph of the function f. If f(x)= k - x2 , where k is a constant, what is the value of k ?
(½)y = 4
x – (½)y= 2
The system of equations above has solution (x, y). What is the value of x ?
(½)y = 4
x – (½)y= 2
The system of equations above has solution (x, y). What is the value of x ?
A landscaper is designing a rectangular garden. The length of the garden is to be 5 feet longer than the width. If the area of the garden will be 104 square feet, what will be the length, in feet, of the garden?
A landscaper is designing a rectangular garden. The length of the garden is to be 5 feet longer than the width. If the area of the garden will be 104 square feet, what will be the length, in feet, of the garden?
In the xy-plane, the graph of which of the following equations is perpendicular to the graph of the equation above?
The slope-intercept form is one of the most common ways to represent a line's equation. For example, the slope of a straight line, slope-intercept, and y-intercept formula determine the equation of a line (where the line intersects the y-axis at the point of the y-coordinate). An equation must be satisfied by each point on a line. For example, the graph of the linear equation y = mx + c is a line with slope m and y-intercept m and c. This is known as the slope-intercept form of the linear equation, and the values of m and c are real numbers.
¶A line's slope, m, represents its steepness. Sometimes the slope of a line is referred to as the gradient. A line's y-intercept, b, represents the y-coordinate of the point where the line's graph intersects the y-axis.
In the xy-plane, the graph of which of the following equations is perpendicular to the graph of the equation above?
The slope-intercept form is one of the most common ways to represent a line's equation. For example, the slope of a straight line, slope-intercept, and y-intercept formula determine the equation of a line (where the line intersects the y-axis at the point of the y-coordinate). An equation must be satisfied by each point on a line. For example, the graph of the linear equation y = mx + c is a line with slope m and y-intercept m and c. This is known as the slope-intercept form of the linear equation, and the values of m and c are real numbers.
¶A line's slope, m, represents its steepness. Sometimes the slope of a line is referred to as the gradient. A line's y-intercept, b, represents the y-coordinate of the point where the line's graph intersects the y-axis.
Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 per gallon, which equation can Alan use to determine how many fewer average miles, m, he should drive each week?
To calculate average miles, divide the total distance traveled by the time spent traveling. This will provides us with your average speed.
So, for example, if Ben traveled 150 miles in 3 hours, 120 miles in 2 hours, and 70 miles in an hour, his average speed was about 57 miles per hour. In this case, Alan can travel a hundred miles per week at 25 miles per gallon of gasoline to save $5 per week on gas, assuming gasoline costs $4 per gallon.
Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 per gallon, which equation can Alan use to determine how many fewer average miles, m, he should drive each week?
To calculate average miles, divide the total distance traveled by the time spent traveling. This will provides us with your average speed.
So, for example, if Ben traveled 150 miles in 3 hours, 120 miles in 2 hours, and 70 miles in an hour, his average speed was about 57 miles per hour. In this case, Alan can travel a hundred miles per week at 25 miles per gallon of gasoline to save $5 per week on gas, assuming gasoline costs $4 per gallon.
Point P is the center of the circle in the figure above. What is the value of x ?
Point P is the center of the circle in the figure above. What is the value of x ?
The circle above with center O has a circumference of 36. What is the length of minor arc 
?
The diameter of a circle is also known as its measurement of the circle's edge, circumference, or perimeter.
As opposed to this, a circle's area indicates the space it occupies.
The circle circumference is the length when we cut it, open and draw a straight line from it.
Units like centimeters or meters are typically used to measure it.
The circle's radius is considered when applying the formula to determine the circumference of the circle.
Therefore, to calculate a circle's circumference, we must know its radius or diameter.
Therefore, the circumference of a circle formula is the circle perimeter or circumference is 2πR.
where,
R is the circle's radius.
π is a mathematical constant with an estimated value of 3.14 (to the nearest two decimal places).
The circle above with center O has a circumference of 36. What is the length of minor arc ?
The diameter of a circle is also known as its measurement of the circle's edge, circumference, or perimeter.
As opposed to this, a circle's area indicates the space it occupies.
The circle circumference is the length when we cut it, open and draw a straight line from it.
Units like centimeters or meters are typically used to measure it.
The circle's radius is considered when applying the formula to determine the circumference of the circle.
Therefore, to calculate a circle's circumference, we must know its radius or diameter.
Therefore, the circumference of a circle formula is the circle perimeter or circumference is 2πR.
where,
R is the circle's radius.
π is a mathematical constant with an estimated value of 3.14 (to the nearest two decimal places).
