How Do You Solve Quadratic Equations and Predict Zeroes Without a Calculator?

  • Thread starter Divina
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    Grade 11
So, you need to figure out what to add and subtract, and then you will see that you can make a perfect square from the first three terms.In summary, the conversation discusses the process of finding the a(x-h)²+k form for a quadratic function, without using a graph or calculator. It also explores different aspects of the function, such as the axis of symmetry, maximum or minimum value, vertex coordinates, domain and range, y-intercept, and zeroes. The conversation also includes a discussion on predicting the number of zeroes for a given quadratic function and provides examples to support this prediction. The process of completing the square is mentioned as a method for finding the a(x-h)²+k form.

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  • #1
Divina
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Homework Statement


f(x) : ax²+bx+c, a≠ 0

[have to find the a(x-h)²+k form first]

Find an expression in terms of a, b, c for: (without using graph or calculator)
(i) An equation of the axis of symmetry
(ii) The maximum or minimum value
(iii) The coordinated of the vertex
(iv) The domain and the range
(v) The y-intercept of the graph of the function
(vi) The zeroes of the function

Discuss how you can predict the number of zeroes for a given quadratic function of the form y= ax²+bx+c, a≠ 0. Support the validit of your prediction with some examples.



My attempt was
(ax² + bx + b/2 - b/2) + c
(ax² + bx + b/2) + c - b/2
and after this I'm stuck

(v) y intercept is (0,c)
 
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  • #2
ax2+bx+c=A(x-h)2+k and expand the right side and then equate coefficients if you're unable to do it the other way.

Try adding and subtracting b/4a2 instead of b/2.
 
  • #3
rock.freak667 said:
ax2+bx+c=A(x-h)2+k and expand the right side and then equate coefficients if you're unable to do it the other way.

Try adding and subtracting b/4a2 instead of b/2.

umm thanks:) but i can't make ax2+bx+c=A(x-h)2+k because ax2+bx+c is like an equation just like 3x2-6x+16
and can you please explain me how did you come up with b/4a2
 
  • #4
I meant expand out A(x-h)2 and then equate 'a' to whatever the coefficient of x2 is in the expansion.

Sorry, I was supposed to type b2/4a2. Well remember that you want to have a perfect square (x-h)2 which give x2-2hx+h2 when expanded out.
 
  • #5
Divina said:
umm thanks:) but i can't make ax2+bx+c=A(x-h)2+k because ax2+bx+c is like an equation just like 3x2-6x+16
and can you please explain me how did you come up with b/4a2
No, ax2+bx+c is not "like" an equation; an equation can be recognized by the presence of the = sign. ax2+bx+c is an expression, just as 3x2-6x+16 is.

Since you are NOT working with an equation, you are very limited in the things you can do. You can't add the same amount to both sides, because there are not two sides. About all you can do is add 0.

You want to complete the square by adding a certain amount and subtracting exactly the same amount so that the net change is 0.

For example,
22 + 4x
= 2(x2 + 2x)
= 2(x2 + 2x + 1) - 2
= 2(x + 1)2 - 2

In the 3rd line I added 1 inside the parentheses, but I really added 2, so to keep the expression equal to the previous one, I had to balance things by adding -2 (or subtracting 2).
 

FAQ: How Do You Solve Quadratic Equations and Predict Zeroes Without a Calculator?

What is a quadratic equation?

A quadratic equation is a polynomial equation of the second degree, meaning it has one variable raised to the power of two. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants.

How do you solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. You can also factor the equation or complete the square to find the solutions.

What is the difference between a linear and quadratic equation?

A linear equation has one variable raised to the first power, while a quadratic equation has one variable raised to the second power. Linear equations result in a straight line when graphed, while quadratic equations result in a parabola.

What are the applications of quadratic equations?

Quadratic equations have many real-world applications, such as determining the maximum or minimum value of a function, predicting the trajectory of a projectile, and modeling the growth or decay of populations.

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation is the expression inside the square root in the quadratic formula: b² - 4ac. It can tell you the number and nature of solutions to the equation. A positive discriminant means there are two real solutions, a zero discriminant means there is one real solution, and a negative discriminant means there are no real solutions.

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