To determine which equation can be solved using the given expression [tex]\(\frac{-3 \pm \sqrt{(3)^2+4(10)(2)}}{2(10)}\)[/tex], we will start by recognizing that this expression is derived from the quadratic formula applied to a quadratic equation in the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Given the expression is [tex]\(\frac{-3 \pm \sqrt{(3)^2 + 4(10)(2)}}{2(10)}\)[/tex], we can identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- [tex]\(a = 10\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = 2\)[/tex]
Next, we substitute these coefficients into the standard form of a quadratic equation:
[tex]\[ 10x^2 - 3x + 2 = 0 \][/tex]
This is the equation that corresponds to the given quadratic formula. Now, let's rewrite this equation by isolating the terms in different ways to match the given options.
Starting with:
[tex]\[ 10x^2 - 3x + 2 = 0 \][/tex]
We reorganize and isolate the constant term on one side:
[tex]\[ 10x^2 - 3x = -2 \][/tex]
[tex]\[ 10x^2 - 2 = 3x \][/tex]
All the options presented are:
a) [tex]\( 10x^2 = 3x + 2 \)[/tex]
b) [tex]\( 2 = 3x + 10x^2 \)[/tex]
c) [tex]\( 3x = 10x^2 - 2 \)[/tex]
d) [tex]\( 10x^2 + 2 = -3x \)[/tex]
From our rearrangement, we notice that:
[tex]\[ 10x^2 - 2 = 3x \][/tex]
We can write this equivalently as:
[tex]\[ 10x^2 + 2 = -3x \][/tex]
Therefore, the correct equation that can be solved using the given expression is:
[tex]\[ 10x^2 + 2 = -3x \][/tex]
Thus, the answer is:
[tex]\[ \boxed{10x^2 + 2 = -3x} \][/tex]
