The roots of a quadratic equation can be found by using the quadratic formula: x = (-b ± √(b² – 4ac)) / 2a.
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The famous mathematician, Carl Friedrich Gauss once said, “Mathematics is the queen of the sciences.” It is true that mathematics has played a significant role in many fields of study and has been an essential part of many technological advancements. When it comes to quadratic equations, finding the roots of an equation is an important part of solving real-world problems.
So, how do you find the roots of a quadratic equation? The answer lies in the quadratic formula. The quadratic formula is a formula that can be used to find the roots of any quadratic equation.
The formula is as follows:
x = (-b ± √(b² – 4ac)) / 2a
Where a, b, and c are coefficients of the quadratic equation ax²+bx+c=0. To use this formula, you need to plug in the values of a, b, and c into the formula and solve for x.
Interesting facts about quadratic equations:
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Quadratic equations have been around since ancient times. The ancient Greeks were studying quadratic equations more than 2000 years ago.
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In the ninth century, Indian mathematician Mahavira created a list of quadratic equations with positive integer solutions.
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Quadratic equations are often used in physics to describe the motion of objects under the influence of gravity.
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The discriminant of a quadratic equation can tell you whether the roots of the equation are real, imaginary, or complex.
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If the discriminant is greater than zero, then the roots are real and distinct. If the discriminant is zero, then the roots are real and equal. If the discriminant is less than zero, then the roots are complex conjugates.
Here is a table to illustrate the different types of roots of a quadratic equation:
| Discriminant | Roots of the equation |
|---|---|
| > 0 | Two real and distinct roots |
| = 0 | Two real and equal roots |
| < 0 | Two complex conjugate roots |
In conclusion, finding the roots of a quadratic equation is an essential part of solving real-world problems. The quadratic formula is a formula that can be used to find the roots of any quadratic equation, and understanding the discriminant can help you determine the type of roots the equation has. As Albert Einstein once said, “Pure mathematics is, in its way, the poetry of logical ideas.”
See the answer to “How do you find the roots of a quadratic equation?” in this video
This video teaches how to find the roots of a quadratic equation using the quadratic formula. The first step is to set the quadratic equal to zero and label the coefficients as a, b, and c. Then, we calculate the discriminant to determine the nature of the solutions. If the discriminant is not a square number, we get two real irrational solutions. Finally, we plug in the values of a, b, and c into the quadratic formula and simplify the expression by dividing each term by 2 if possible.
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Important Formulas on Roots of Quadratic Equations: The roots are calculated using the formula, x = (-b ± √ (b2 – 4ac) )/2a. Discriminant is, D = b2 – 4ac. If D > 0, then the equation has two real and distinct roots.
The roots of a quadratic equation ax 2 + bx + c = 0 can be found using the quadratic formula that says x = (-b ± √ (b 2 – 4ac)) /2a. Alternatively, if the quadratic expression is factorable, then we can factor it and set the factors to zero to find the roots.
We use the quadratic formula to find the roots of a quadratic equation. The formula is given as x = − b ± b 2 − 4 a c 2 a Here b 2 – 4ac is called the discriminant. It is denoted by D.
Method 1: The roots of the quadratic equations can be found by the Shridharacharaya formula. x = [-b±√(b 2 – 4ac)]/2a Example: The length of sides of a rectangle is given by x – 3 and x – 5 and the area of the rectangle is 3 unit 2 .
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The general form of a quadratic equation is given by [math]ax^2+bx+c=0\tag*{}[/math]
• Geometrically it represents a parabola
• If [math]a%3E0[/math], this parabola opens upwards and we say that this has a concave up shape
• If [math]a%3C0[/math], this parabola opens downwards and we say that this has a concave down shape
• If [math]a=0[/math], this becomes a straight line
• Roots of the quadratic equation is the [math]x[/math] intercepts (if exists) of the parabola.
• The roots of a quadratic equation can be found by factoring
• It can be found by using quadratic formulaDerivation of the quadratic formula:
We simply require to use the completing square technique…
[math]\begin{equation}\begin{split}ax^2+bx+c&=0\a\left(x^2+\dfrac bax
ight)+c&=0\a\left(x^2+2\cdot x\dfrac b{2a}+\dfrac{b^2}{4a^2}-\dfrac{b^2}{4a^2}
ight)+c&=0\a\left(x+\dfrac b{2a}
ight)^2-\dfrac{b^2}{4a}+c&=0\a\left(x+\dfrac b{2a}
ight)^2&=\dfrac{b^2}{4a}-c\\left(x+\dfrac b{2a}
ight)^2&=\dfrac{b^2-4ac}{4a^2}\x+\df…
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Herein, How do you find the root of a quadratic function? As an answer to this: All you need to do then is do the opposite of b so since b is negative five b is our opposite of b is going to be positive. Five. So x equals.
Similarly, How do you find the roots of a quadratic equation quickly?
The reply will be: And let’s solve the given equation. At first we will multiply the coefficient of x square. And the constant. So 2 into 2 results for now.
What are the 4 ways to find the roots of quadratic equation?
Response: Answer: There are various methods by which you can solve a quadratic equation such as: factorization, completing the square, quadratic formula, and graphing. These are the four general methods by which we can solve a quadratic equation.
Similarly one may ask, What are the 3 ways to find roots of a quadratic? Answer to this: There are three basic methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square.
Accordingly, How do you find the roots of a quadratic function? Another way to find the roots of a quadratic function. This is an easy method that anyone can use. It is just a formula you can fill in that gives you roots. The formula is as follows for a quadratic function ax^2 + bx + c: These formulas give both roots. When only one root exists, both formulas will give the same answer.
Also Know, What are the roots of the quadratic equation x 2 – 7x + 10 = 0? Answer will be: For example, the roots of the quadratic equation x 2 – 7x + 10 = 0 are x = 2 and x = 5 because they satisfy the equation. i.e., when each of them is substituted in the given equation we get 0. when x = 2, 2 2 – 7 (2) + 10 = 4 – 14 + 10 = 0. when x = 5, 5 2 – 7 (5) + 10 = 25 – 35 + 10 = 0.
In this way, How do you write a quadratic equation? Here’s how you do it: Write down the quadratic formula. The quadratic formula is: Identify the values of a, b, and c in the quadratic equation. The variable a is the coefficient of the x 2 term, b is the coefficient of the x term, and c is the constant. For the equation 3x 2 -5x – 8 = 0, a = 3, b = -5, and c = -8.
Also asked, How do you solve a square root equation? However, unlike the other operations, when we take the square root we must remember to take both the positive and the negative square roots. Now solve a few similar equations on your own. Solve x^2=16 x2 = 16. Solve x^2=81 x2 = 81. Solve x^2=5 x2 = 5. x, equals, plus minus, square root of, 5, end square root
Likewise, How do you find the quadratic root of X? Answer will be: Here, a = 1, b = -7 and c = 10. Then by quadratic formula: Therefore, x = 2, x = 5. Complete the square on the left side. Solve by taking square root on both sides. Example: Find the quadratic roots of x 2 – 7x + 10 = 0 by completing square. By completing the square, we get (x – (7/2) ) 2 = 9/4.
Also, Can a quadratic equation be solved by taking the square root?
Answer to this: Not all quadratic equations are solved by immediately taking the square root. Sometimes we have to isolate the squared term before taking its root. For example, to solve the equation 2x^2+3=131 2×2 +3 = 131 we should first isolate x^2 x2. We do this exactly as we would isolate the x x term in a linear equation.
How do you calculate the roots of an equation? The roots are calculated using the formula, x = (-b ± √ (b 2 – 4ac) )/2a. Discriminant is, D = b 2 – 4ac. If D > 0, then the equation has two real and distinct roots. If D < 0, the equation has two complex roots. If D = 0, the equation has only one real root.
Also asked, What are the roots of a quadratic equation Ax 2 bx + c 0? For a given quadratic equation ax 2 + bx + c = 0, the values of x that satisfy the equation are known as its roots. i.e., they are the values of the variable (x) which satisfies the equation. The roots of a quadratic function are the x-coordinates of the x-intercepts of the function.