Find the real solutions of the equation by graphing. 6x^3-2x=-9x^2
Graph a, the solution is 3.
To find the real solutions of the equation 6x^3 - 2x = -9x^2 by graphing, follow these steps:
Step 1: Rewrite the equation in the form of a polynomial equation equal to zero:
6x^3 - 2x + 9x^2 = 0
Step 2: Plot the graph of the equation by identifying the x-intercepts, where the graph intersects the x-axis. To do this, set the equation equal to zero and solve for x:
6x^3 - 2x + 9x^2 = 0
Step 3: Factoring is the best approach to solve this equation. However, if it seems difficult to factorize the equation, you can use a graphing calculator or software to plot the graph.
Step 4: Observe the graph and identify the points where the graph intersects the x-axis. These are the x-values representing the real solutions to the equation.
Alternatively, you can use technology like a graphing calculator or software such as Desmos or Wolfram Alpha to quickly graph the equation and see the x-intercepts.
To find the real solutions of the equation 6x^3 - 2x = -9x^2 by graphing, follow these steps:
Step 1: Rearrange the equation to bring all terms to one side of the equation, making it equal to zero:
6x^3 - 9x^2 - 2x = 0
Step 2: Plot the equation on a coordinate plane. Choose a range of values for x and calculate the corresponding y-values using the equation.
Step 3: Connect the points together to form a smooth curve that represents the graph of the equation.
Step 4: Examine the x-axis for the points where the graph intersects it. These points represent the real solutions of the equation.
However, keep in mind that actually graphing the equation may not always be the most efficient method. In this case, a more algebraic approach can help find the solutions of the equation faster.
Let's solve the equation using an algebraic method:
6x^3 - 2x = -9x^2
Step 1: Rearrange the equation to bring all terms to one side of the equation:
6x^3 + 9x^2 - 2x = 0
Step 2: Factor out any common terms if possible. In this case, there are no common factors.
Step 3: Look for possible rational roots using the Rational Root Theorem. The possible rational roots of 6x^3 + 9x^2 - 2x = 0 can be found by taking the factors of the constant term (in this case, 0) and dividing them by the factors of the leading coefficient (in this case, 6).
Possible rational roots are ±(factors of 0) / (factors of 6), which simplifies to just ±(factors of 6).
The factors of 0 are 1, -1, 2, -2, 3, -3, 6, and -6.
The factors of 6 are 1, -1, 2, -2, 3, -3, 6, and -6.
So, the possible rational roots are ±1, ±2, ±3, and ±6. These are the values to test.
Step 4: Use synthetic division to test each potential rational root. Start with the first one, plug it into the equation, and see if it results in a remainder of zero.
For example, let's test x = 1:
1 | 6 9 -2 0
---
6 15 13 13
The remainder is 13, which is not zero. Therefore, 1 is not a root of the equation.
Continue testing the other possible rational roots until you find one that results in a remainder of zero.
By applying the synthetic division with x = 3, we find:
3 | 6 9 -2 0
---
18 81 235 705
The remainder is zero, indicating that x = 3 is a root of the equation.
Step 5: Use polynomial long division to divide the original equation by the root found in Step 4 (x = 3). This will give you a quadratic equation that can be solved using factoring or the quadratic formula.
Using polynomial long division, we get:
(6x^3 + 9x^2 - 2x) ÷ (x - 3) = 6x^2 + 27x + 79
Now, 6x^2 + 27x + 79 = 0 can be solved using factoring or the quadratic formula.
After finding the roots, you can substitute them back into the original equation to ensure they are solutions.
In summary, while graphing can provide a visual representation of the equation and its solutions, it may not always be the most efficient method. An algebraic approach, such as factoring or using the quadratic formula, can often yield results more quickly.
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