Algebra I

posted by .

Can anyone summarize the quadratic formula? I am trying to understand this for algebra without any unnecessary and overused words involved like Wikipedia and sites that give me a headache. I just want a paragraph (or more than if its impossible to paraphrase) that is straight, specific and to the point. Thanks in advance.

• Algebra I -

Factoring

By far the simplest way of solving quadratic equations is by direct factoring. This does, however, depend on the ability to visualize the exact terms of the factors and definitely improves with experience. Take the following example for instance:
1--Given x^2 + (6/3)x - (35/3) = 0
2--Multiplying through by 3 gives us 3x^2 + 16x - 35 = 0
3--We know that the factors take the form of (ax +/-b)x(cx +/-d)
4--Therefore, we must find values of a, b, c, and d that satisfy ax(x) = 3x^2, ax(+/-d) + cx(+/-b) = +16 and (+/-b)
(+/-d) = -35.
5--Clearly, ether a or c = 3 and b(d) = -5(7) or 5(-7)
6--A little mental arithmetic leads us to a = 3, b = -5, c = 1, and d = +7
7--This leaves us with (3x - 5)(x + 7) = 0
8--If either (3x - 5) or (x + 7) is zero, their product is zero
9--Therefore, 3x - 5 = 0 and x + 7 = 0 making x = +5/3 or -7.

Completing the Square

This method depends on the simplification of the quadratic equation by adding an expression to both sides of the equation that makes one side a perfect square. The process involves the following steps:
1--Simplify and rearrange the equation such that the x^2 and x terms are all on one sides of the equation.
2--Force the coefficient of x^2 to be unity and positive by dividing through by the appropriately signed coeffiecient of x^2.
3--Add the square of half the coefficient of x to both sides of the equation.
4--Take the square root of both sides.
5--Solve the resulting simplified equations.

An example will illustrate the process.

1--Given x^2 - 6x - 16
2--Rearranging, x^2 - 6x = 16
3--Adding (6/2)^2 to both sides gives x^2 - 6x + 9 = 16 + 9 = 25
4--By inspection, x^2 - 6x + 9 = (x - 3)^2 = 25
5--Taking the square root of both sides, (x - 3) = +/-5
6--Therefore, x = 3 + 5 = 8 or x = 3 - 5 = -2.

1--Given 3x^2 = 32 - 10x.
2--Rearranging, 3x^2 + 10x - 32.
3--Dividing through by 3 gives x^2 + (10/3)x = 32/3.
4--Adding (10/3)/2 to both sides gives x^2 + (10/3)x + (5/3)^2 = 32/3 + (5/3)^2 = 121/9.
5--By inspection, x^2 + (10/3)x + (5/3)^2 = (x + (5/3))^2 = 121/9.
6--Taking the square root of both sides, [x + (5/3)] = +/-11/3.
7--Therefore, x = -5/3 + 11/3 = 2 or x = -5/3 - 11/3 = -5 1/3.

Quadratic equations are typically solved by simple factoring or the quadratic formula, x = [-b+/-sqrt(b^2 - 4ac)]/2a. Every quadratic equation can be written in the form ax^2 + bx + c = 0, where a, b, c, may have any numerical values. If we can solve this quadratic, we can solve any quadratic equation.
1--Transposing, we have ax^2 + bx = -c
2--Dividing both sides by a we have x^2 + bx/a = -c/a
3--Adding (b/2a)^2 to each side we have x^2 + bx/a + (b/2a)^2 = b^2/4a^2 - c/a
4--Simplifying we have (x + b/2a)^2 = (b^2 - 4ac)/4a^2
5--Extracting the sqrt we have x + b/2a = +/-[sqrt(b^2 - 4ac)]/2a
6--Therefore, the final quadratic formula becomes
...................x = [-b +/-sqrt(b^2 - 4ac)]
......................................2a

Variations of the quadratic formula are

...................x = -(b/2a) +/-sqrt[(b/2a)^2 - c/a] and

...................x = sqrt[(b/2)^2 - ac] - (b/2)
......................................a

Example: Solve 2X^2 - 38X + 96 = 0.
1--a = 2, b = -38, and c = 96.
2--x = { -(-38) +/- sqrt[(-38^2) - 4(2)(96)]}/2(2)
.......= {38 +/- sqrt[(1444) - 768]}/4
.......= {38 +/- sqrt[676]}/4
.......= {38 +/- 26}/4
.......= {38 + 26}/4 or {38 - 26}/4
.......= 64/4 and 12/4
.....x = 16 and 3.

There is another way to solve quadratics that, in many instances, is just as expedient, if not often simpler. The method requires that the given expression be modified to the form of (mx +/- n)^2 = p giving us solutions of x = (p + n)/m and (p - n)/m. An example will help to visualize the process.

1--Given an expression of the form ax^2 + bx + c = 0 such as x^2 - 10x + 16 = 0.
2--Multiply the expression by a number q that results in q(a) being a perfect square and q(b) being evenly divisible by 2sqrt[q(a)].
3--The number 4 fits our need here resulting in 4x^2 - 40x + 64 = 0.
4--By inspection, we see that 4x^2 - 40x derives from [(sqrt(qa))x - (qb/2sqrt(qa))]^2 or (2x - 10)^2 which results in 4x^2 - 40x + 100.
5--Adding 36 to the right side gives us 4x^2 - 40x + 100 = 36 thereby retaining our original expanded equality.
6--Thus, we end up with (2x - 10)^2 = 36 or (2x - 10) = +6 or -6.
7--Therefore, x = +16/2 = 8 or x = +4/2 = 2.

1--Given 3x^2 - 29x + 154 = 0
2--Multiplying through by 12 yields 36x^2 - 1044x + 5544 = 0
3--By inspection, we see that 36x^2 - 1044x derives from (6x - 87)^2 which gives us 36x^2 - 1044x + 7569.
4--Adding 2025 to the right side gives us 36x^2 - 1044x + 7569 = 2025 thereby retaining our original expanded equality.
5--This leads us to (6x - 87)^2 = 2025 or (6x - 87) = +/-45
6--Then, x = +132/6 = 22 or +42/6 = 7.

Give it a try the next time you are confronted with a quadratic equation to solve. With practice and expreience, it might be just as quick as he quadratic formula.

• Algebra I -

7x-y+5

Similar Questions

1. Math

Use the quadratic formula to solve each of the following quadratic equations... 1. 2x^2-5x=3 2. 3x^2-2x+1=0 Rearrange the equation in quadratic formula form. 2x^2 -5x -3 = 0 Then use the formula. Tell me what you don't understand about …

I am trying to define the different appraoches to solving quadratic equations. My book says using quadratic formula, completing the sqaure and factoring. I thought completing the square would be by facotring?
3. math

ok so I asked a question like this before and I understood how to do it and a cuple more like it but now I don't understand how to do this on ..... I keep on trying to find the answer but I just don't understand what I am supposed …

You know how to solve quadratic equations using algebra, graphs and the quadratic formula. Sometimes one method of solving is more convenient than another method. Describe how you would solve each equation. Give reasons for you answers. …
5. Algebra

can show me how to do this with out just squaring 3x^(1/2) - x - 6 = 0 how do i do this and solve for x with out just squareing and using quadratic formula. I was never taught the proof of the quadratic formula and would like to know …
6. english

Share your thoughts on the following statement about writing from William Strunk and E. B. White's little book The Elements of Style. Think of one or two ways the writing you have worked on and revised might be like a drawing or a …
7. Algebra

I was given this answers to a problems I solved but I do not understand what she means hope you can help. Before attempting to solve this quadratic equation, determine how many solutions there will be for this quadratic equation. Explain …
8. Algebra

I was given this answers to a problems I solved but I do not understand what she means hope you can help. Before attempting to solve this quadratic equation, determine how many solutions there will be for this quadratic equation. Explain …
9. Science

Please summarize this. I got it from a website and im doing my notecards. Im not trying to plagiarize, Im just not a science person. Please just summarize it so I can just understand it so I can explain it in my own words: The light …
10. Language arts 1 question please

edgar allen poe uses a pair of rhyming words to describe trying to hold sand in "a dream within a dream". in a short paragraph tell what the words are and how they express the speakers feelings I do NOT want someone to give me the …

More Similar Questions