The tangent to the curve $$ y=x^{2}-5 x+3 $$ at a certain point is parallel to the line $$ y=x $$. Find its equation and where it cuts the $$ x $$-axis. olution:

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To find the equation of the tangent to the curve $$ y = x^{2} - 5x + 3 $$ that is parallel to the line $$ y = x $$ and determine where this tangent line intersects the $$ x $$-axis, follow these steps:

### 1. **Determine the Slope of the Tangent Line**

The given line $$ y = x $$ has a slope $$ m = 1 $$. Since the tangent line is parallel to this line, it must also have the same slope.

### 2. **Find the Derivative of the Curve**

The derivative of the curve $$ y = x^{2} - 5x + 3 $$ with respect to $$ x $$ gives the slope of the tangent at any point $$ x $$ on the curve:

$$
\frac{dy}{dx} = 2x - 5
$$

Set this equal to the slope of the parallel line:

$$
2x - 5 = 1
$$

### 3. **Solve for $$ x $$**

$$
2x - 5 = 1 \
2x = 6 \
x = 3
$$

### 4. **Find the Corresponding $$ y $$-Coordinate**

Substitute $$ x = 3 $$ back into the original equation to find $$ y $$:

$$
y = (3)^{2} - 5(3) + 3 = 9 - 15 + 3 = -3
$$

So, the point of tangency is $$ (3, -3) $$.

### 5. **Write the Equation of the Tangent Line**

Using the point-slope form of a line:

$$
y - y_{1} = m(x - x_{1})
$$

Substitute $$ m = 1 $$ and $$ (x_{1}, y_{1}) = (3, -3) $$:

$$
y - (-3) = 1(x - 3) \
y + 3 = x - 3 \
y = x - 6
$$

### 6. **Find Where the Tangent Line Intersects the $$ x $$-Axis**

To find the $$ x $$-intercept, set $$ y = 0 $$ and solve for $$ x $$:

$$
0 = x - 6 \
x = 6
$$

So, the tangent line intersects the $$ x $$-axis at the point $$ (6, 0) $$.

### **Summary**

- **Equation of the tangent line:** $$ y = x - 6 $$
- **Point where the tangent intersects the $$ x $$-axis:** $$ (6, 0) $$
The tangent line to the curve $$ y = x^{2} - 5x + 3 $$ that is parallel to $$ y = x $$ is $$ y = x - 6 $$. This tangent intersects the $$ x $$-axis at the point $$ (6, 0) $$.

The tangent to the curve at a certain point is parallel to the line .
Find its equation and where it cuts the -axis.
olution:

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The tangent to the curve at a certain point is parallel to the line . Find its equation and where it cuts the -axis. olution: