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2 years ago

How many solutions are there to this equation 3x(x-4)+5-x=2x-7

Mathematics

1 answer:

stich3 [128]2 years ago

4 0

Answer:

2 solutions

Step-by-step explanation:

I like to use a graphing calculator to find solutions for equations like these. The two solutions are ...

x = 1
x = 4

To solve this algebraically, it is convenient to subtract 2x-7 from both sides of the equation:

3x(x -4) +5 -x -(2x -7) = 0

3x^2 -12x +5 -x -2x +7 = 0 . . . . . eliminate parentheses

3x^2 -15x +12 = 0 . . . . . . . . . . . . collect terms

3(x -1)(x -4) = 0 . . . . . . . . . . . . . . . factor

The values of x that make these factors zero are x=1 and x=4. These are the solutions to the equation. There are two solutions.

Alternate method

Once you get to the quadratic form, you can find the number of solutions without actually finding the solutions. The discriminant is ...

d = b^2 -4ac . . . . where a, b, c are the coefficients in the form ax^2+bx+c

d = (-15)^2 -4(3)(12) = 225 -144 = 81

This positive value means the equation has 2 real solutions.

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What much expression represents a rational number?

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Answer:

$\displaystyle \frac{2}{7}+\sqrt{121}$

Step-by-step explanation:

Rational Numbers

A rational number is any number that can be expressed as a fraction

$\displaystyle \frac{a}{b}, \ b\neq 0$

for a and b any integer and b different from 0.

As a consequence, any number that cannot be expressed as a fraction or rational number is defined as an Irrational number.

Let's analyze each one of the given options

$\displaystyle \frac{5}{9}+\sqrt{18}$

The first part of the number is indeed a rational number, but the second part is a square root whose result cannot be expressed as a rational, thus the number is not rational

$\pi + \sqrt{16}$

The second part is an exact square root (resulting 4) but the first part is a known irrational number called pi. It's not possible to express pi as a fraction, thus the number is irrational

$\displaystyle \frac{2}{7}+\sqrt{121}$

The square root of 121 is 11. It makes the whole number a sum of a rational number plus an integer, thus the given number is rational

$\displaystyle \frac{3}{10}+\sqrt{11}$